## 1 Introduction

### 1.1 General Background

This chapter discusses self-assembly of polymer solutions and solutions of polymer mixtures under external fields, specifically under shear fields, as observed by real-time and in situ light scattering (LS). Although shear-induced structures of solutions of polymer mixtures will be discussed in some parts of this chapter, the discussion can be in principle extended to shear-induced structures of binary polymer mixtures in bulk. This is because the solvent used is a neutral solvent which effectively weakens segmental interactions of polymers as will be detailed later (*Section*
*5.1*
).

Our research theme to be treated here is considered to belong to the general theme on pattern formation in nature. This is because the pattern formation in various systems in nature occurs under the influence of external fields and is inherently related to the self-assembly in the so-called *open nonequilibrium systems*, i.e., systems that are open to various external energy flows and are brought into nonequilibrium states. *Self-assembly* is defined here as a term which includes self-assembling processes, mechanisms, and structures of molecules. It is possible to consider these systems as a model of *soft matters* (or *complex liquids*) that are composed of matters such as polymers, liquid crystals, membranes, gels, micelles, colloids, supercooled liquids, multi-component liquids, etc. They are assemblies of molecules and/or supramolecules which commonly have characteristic structures and properties in between solids and liquids, depending on a space–time scale of observation.

*dissipative structures*, which are defined as ordered structures developed in open nonequilibrium systems. How can the dissipative structures be observed?

*Figure*

*8-1*shows a general principle of the method. Energy (e.g., strain) is imposed on the systems and the response (e.g., stress) to the input energy is observed. Simultaneously an incident beam (e.g., light, X-ray, neutron) is sent to the systems to monitor structural responses within the system to the applied energy over a wide length scale. Hopefully, correlation can occur between the physical properties (i.e., response to input energies) and structural responses elucidated by in situ, simultaneous scattering experiments with the guide of statistical mechanical theories, as shown by the broken arrow in the figure. As for applied fields, steady, oscillatory, and step-up or step-down shear flow, etc. will be discussed.

### 1.2 Principles of Rheo-Optics

*rheo-optical methods*will be applied to explore the dissipative structures. The principle of rheo-optics is illustrated in

*Figure*

*8-2*. Rheo-optical methods involve simultaneous measurements of both rheological and optical properties of systems to clarify their complex rheological properties on the basis of responses of fundamental structural units of the systems over a range of length scale as wide as possible (e.g., ∼0.1 nm–∼10 μm).

Upon applying mechanical excitations to these systems, mechanical responses are detected and analyzed. *Rheology* clarifies the law of causality between mechanical stimuli and responses. Simultaneously optical responses or responses of structure factors are detected and analyzed on the basis of principles established in *optics*. The analysis of optical responses will help developing nonequilibrium statistical mechanical theories to predict the free energy associated with the applied mechanical excitations. The prediction is then fed back to predict further mechanical responses associated with the particular optical responses analyzed by the rheo-optical methods. Thus rheo-optics is anticipated to give directly the molecular or structural origin of rheological properties beyond phenomenological or mathematical understanding of the properties. As for optical responses, one can think of optical images such as those obtained by transmission and reflection optical microscopy, fluorescence and Raman microscopy, phase-contrast microscopy, and polarizing light microscopy. One can think of various scattering methods such as light, X-ray, and neutron scattering as well as spectroscopy such as Raman and infrared spectroscopy. Birefringence and optical dichroism are also useful methods.

## 2 Shear Rheo-Optics

### 2.1 Background of Shear Rheo-Optics

*Figure*

*8-3*represents a schematic illustration of the rheo-optical apparatus under shear deformation and/or flow. In this apparatus, a rheometer to measure strain–stress behaviors is linked with a transmission light microscope (OM) and a laser LS apparatus. Samples are placed in a shear cell composed of two parallel plates or a set of cone and plate which are transparent for the incident beam, transmitted beam, and the scattered beam. In this figure shear strain is applied by rotating the lower part of the cell (either a plate or cone); shear force and normal force from the sample are measured by a stress transducer fixed to the upper part of the cell. The shear (or flow) direction, strain (or velocity) gradient direction, and neutral (or vorticity) direction are conventionally taken, respectively, along the

*x-*,

*y-*, and

*z*-axis in Cartesian coordinate shown in the figure. In this set up, the OM images and LS patterns are observed in situ and

*at*real-time in the plane parallel to the

*x*-

*z*plane as a function of time

*t*and shear rates \(\dot \gamma\) after imposing a given \(\dot \gamma\) as well as a function of variables inherent in systems under consideration, e.g., temperature, etc. They are observed also as a function of angular frequency ω, amplitude of strain γ

_{0}, strain phase

*ϕ*

_{s}, and time after imposing oscillatory shear. It should be pointed out that Wu et al. designed a shear-LS apparatus which allows the measurement of the scattering in the

*x-y*plane parallel to which the velocity gradient exists [1].

*Figure*

*8-4*[2]. Basic fundamental quantities underlying systems of interest are (1) a spectrum of concentration fluctuations in the case of multicomponent systems or density fluctuations in the case of single component systems, and (2) orientation and deformation of molecules and structural units forming hierarchical structures in general. The scattering and spectroscopy directly explore these quantities under shear. Statistical mechanical studies of rheological behaviours of such structured systems as described above provide challenging fundamental research topics.

*The shear flow may affect the phase transition of mixtures.* Consider a mixture of simple liquids for which the concentration fluctuation is an order parameter. If \(\dot \gamma\) is smaller than the relaxation rate of the concentration fluctuations *Γ*
_{conc}, the concentration fluctuation thermally relaxes before it is deformed under the shear. In this case the shear does not significantly affect the concentration fluctuations and hence the state of the mixtures. However if \(\dot \gamma > \Gamma _{\rm conc}\), the fluctuations cannot thermally relax in the time scale of the given applied field and are hence affected by shear; consequently the shear affects the state of mixtures and hence phase transition. In fact decrease of the critical temperature under shear flow was reported for some mixtures as will be detailed in *Section*
*5.4*
.

*δϕ*(

**r**,

*t*) and velocity field

**v**(

**r**,

*t*) where

*δϕ*is local concentration fluctuation of one of the components in a binary mixture, and

**v**is the average velocity of the system. The time-evolutions of

*δϕ*and

**v**are given in general by coupled nonlinear equations in the context of the Helfand–Fredrickson–Milner–Onuki (HFMO) [3, 4, 5] theory,

*μ*

_{0}is the thermodynamic driving force to create the concentration fluctuations),

*μ*

_{el}is the elastic part of the chemical potential,

*θ*

_{t}and ζ are the random thermal force contributing to

*δϕ*and

**v**, respectively,

*ρ*is the density,

*p*is pressure,

*ϕ*

_{0}is average concentration of one polymer component in binary mixtures, \((\delta F/\delta \phi )_0\) is the variational derivative of free energy functional \(F\left\{ {\delta \phi } \right\}\) of the mixture at

*ϕ*

_{0}, \(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}}\over \sigma }\) is stress tensor, and

*η*

_{s}is the viscosity. In the above equation the elastic effect due to the deformation of polymers affects the \(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}}\over \sigma }\) and

*μ*

_{el}which depend on \(\dot \gamma\). The effect is generally important when \(\dot \gamma\) is faster than the relaxation rate

*Γ*

_{d}of deformation of polymer chains so that the deformations cannot be relaxed under the external field.

*u*

_{r}and

*v*

_{r}are coupled with diffusion

Here *D*
_{ur} and *D*
_{vr} are the diffusion constants of the two species. The reaction-diffusion systems have been found to give various static ordered structures known as *touring patterns* [7] and dynamic ordered patterns known as *Belouzov–Zhabotinksy reaction* [8]. The dissipative structures that have been reported so far are still limited to those in the macroscopic scale. Later it will be shown that polymer solutions and solutions of polymer mixtures under shear flow will give rise to some dissipative structures at mesoscopic length scales.

### 2.2 Shear-Induced Phase Transition: Two Opposing Phenomena, Mixing and Demixing

The shear flow effect on phase transitions was for the first time reported by Silberberg and Kuhn [9] more than a half-century ago. They reported shear-induced mixing of the phase-separated polymer solution composed of a polymer mixture in a solvent. Since then there has been a considerable number of reports as detailed later in *Section*
*5*
and
*6*
.

A series of studies of shear-flow effects on various systems revealed that shear flow changes the equilibrium state of systems: In one case, shear flow brings two-phase systems into a single-phase (*shear-induced mixing or single-phase formation*), but in other cases shear flow brings single-phase systems into two phases (*shear-induced demixing or phase separation*); interestingly enough, these two effects are quite opposite.

*Figure*

*8-5*demonstrates the opposing effects of shear flow as observed by changes in steady-state light scattering patterns, which were observed in the

*x-z*plane where the flow direction (

*x*-axis) is set to vertical. The left half (parts (a) and (b)) shows a disappearance of strong LS with increasing \(\dot \gamma\) from 40 to 250 s

^{−1}at 68°C (8°C below the cloud point), while the right half (parts (c) and (d) shows an appearance of strong LS scattering with increasing \(\dot \gamma\) from 0.4 to 6.3 s

^{−1}at 21°C.

The former system is composed of a 50/50 wt%/wt% mixture of polystyrene (PS) having average molecular weight *M*
_{w} = 2.1 × 10^{5} and polybutadiene (PB) having *M*
_{w} = 3.1 × 10^{5} which are dissolved in an approximately neutral solvent of dioctylphthalate (DOP) with a total polymer concentration of 3.3 wt% where two polymers are weakly overlapping both in a two-phase solution and in a single-phase solution. The system is hereafter designated PS/PB(50:50)/DOP 3.3 wt%. The latter system is composed of a very high molecular weight PS with *M*
_{w} ≅ 1.0 × 10^{7} dissolved in DOP with polymer concentration, *c*, of 2.0 wt% (designated PS1000/DOP 2.0 wt%) where the PS chains are highly entangled with *c*/*c** = 20 where *c** is the overlap concentration of the polymer [10].

In *Figure*
*8-5b*
and
*c*
the bright halo around the dark circle in the center, which corresponds to an incident-beam stopper, is a part of the incident beam and does not represent a true small-angle scattering. A quantitative detection of a weak scattering intensity in the dark region of the pattern shows that the system under the shear flow is in a single-phase solution as will be detailed later. The strong scattering in part (a) represents highly elongated domains designated as *strings*, as will be also detailed later in *Section*
*5.3*
, which are domains rich in one of the polymer solutions in the matrix of a solution rich in the other polymer. The disappearance of the domains is then indicative of the solution being brought into a single phase state under the given shear rate.

On the other hand in the PS/DOP system, a strong scattered intensity as high as 100 times of the scattered intensity from the single-phase solution appears along the flow direction when an imposed shear rate exceeds a critical shear rate. The strong scattered patterns called as *butterfly scattering pattern* reveals that the shear induces strong concentration fluctuations or even phase separation for the single phase solution in a quiescent state.

What are the physical factors which give rise to the quite contrasting effects of shear-induced single phase formation or shear-induced phase separation? A series of studies by the author and coworkers [11] proposes that the *shear-induced single-phase formation* occurs in *dynamically symmetric systems* with sufficiently weak interfacial tension, while the *shear-induced phase separation* occurs in *dynamically asymmetric systems*. Each of the two cases will be discussed in the following section (*Section*
*3*
).

## 3 Dynamical Asymmetry and Stress–Diffusion Coupling in Multicomponent Systems

### 3.1 Dynamical Asymmetry Versus Dynamical Symmetry

*dynamically symmetric mixtures*, component molecules A and B have equal self-diffusion coefficients. Many small-molecule systems belong to this family (the right half of

*Figure*

*8-6a*). Hence the nonequilibrium dynamics and processes for this family have been relatively well-explored. Polymer pairs having about equal degree of polymerization (DP) and monomeric frictional coefficients also belong to this family (the left half of

*Figure*

*8-6a*).

In *dynamically asymmetric mixtures*, component molecules have different mobilities (self-diffusivities) as in the case of mixtures of polymers having different molecular weights, an extreme case of which is obtained for a high-molecular weight polymer in a solvent as schematically shown in the left part of *Figure*
*8-6b*
, or a dispersion of colloidal particles in a solvent, as shown in the right part of *Figure*
*8-6b*
. Although the particles themselves are not molecules, they are considered as one of big spherical supramolecules. The dynamically asymmetric systems have been relatively less explored than their counterparts, despite of the fact that the systems should be more frequently found in nature and hence more general.

Structure formation involves growth of spatial composition fluctuations, which in turn involves naturally local stress and stress relaxation. In the dynamically symmetric systems, this stress is equally divided into the two components, and the stress relaxes at equal rates in the course of structure growth. The stress relaxation rate is expected to be faster than the growth rate of the structure. Consequently the stress does not play important roles on the structure growth. However, in the dynamically asymmetric systems, the component having larger mobility relaxes faster than its counterpart, so that the local stress is primarily borne by the smaller mobility component. This “imbalancely” borne local stress and stress relaxation affect the free energy functional and hence cooperative diffusivity of the slower mobility component of the systems, which should be reflected back to the dynamics of the composition fluctuations and the form of growing patterns in the systems, as summarized in the bottom part of *Figure*
*8-6*
.

This *stress–diffusion* coupling inherent in the asymmetric systems is anticipated to give intriguing effects on the dynamics and pattern formation, theories of which have been pioneered by Onuki [12] for various systems. It is believed that various pattern formations in nature, including various materials and various systems of living bodies, may involve the stress–diffusion coupling. The unique pattern formation due to the effects is expected to be one of the most important future problems to be unveiled, both experimentally and theoretically. Some examples of the unique pattern formation are presented below.

### 3.2 Some Anticipated Effects of Dynamical Asymmetry on Self-Assembly in the Quiescent State

*Figure*

*8-7*shows void formation of charged colloidal particles dispersed in water as observed by laser scanning confocal microscopy (LSCM) [13]. The system states were explored in the parameter space of salt concentration, charge density of the particles, and particle concentration. The void formation was discovered under a special condition of a very small or almost zero salt concentration, a high charge density (e.g., 4.8

*μ*C/cm

^{2}per particle or the effective charge number of 1.4 × 10

^{3}/particle), and a small particle concentration (e.g., 0.1 wt%). The density of the particles was carefully matched to water to avoid gravitational effects on the pattern formation. The average particle diameter is 0.12 μm, and they appear as bright dots in the sliced images of depth resolution of 0.5 μm, because of scattering from the particles in liquid-like state.

The series of patterns were obtained at 2 months after homogenization of the dispersion via vigorous shaking of the cell. These sliced images were taken in the same area of the sample but by changing the slicing position along the incident beam axis, i.e., the depth direction of the dispersion. Statistically identical patterns are observed everywhere in the sample cell, indicating that the pattern was not formed by gravitational effects but rather by long range electrostatic interactions. The patterns are composed of regions rich in the particles, where the particles are in the liquid state, and voids where almost no particles are observed. A careful tracing of the series of the sliced images reveals that the voids are not isolated but are interconnected along the depth and hence cocontinuous with the network-type particle-rich regions. For example, two voids A and B in slice (d) appear to be separated, but they are interconnected in the focal planes located at 6 μm (slice (c)) and 13 μm (slice (b)) above the slice (d).

It is quite striking to note that such a very dilute charged colloidal dispersion forms a 3D continuous network structure rich in particles in the matrix of the void phase under the field of long-range electrostatic interactions, regardless of the origin of the interactions, either DLVO-type repulsive interactions [14, 15] or net attractive interactions [13].

*Figure*

*8-8a*later) or clusters of the particles isolated in water (see

*Figure*

*8-8c*later). Even in the case when the net attractive interactions exist, formation of the isolated clusters as in

*Figure*

*8-8c*, seems to be more reasonable than the network formation.

*Figure*
*8-8*
schematically shows a model for the pattern formation [11]. When the systems having a uniform particle distribution (part a) start to build up concentration fluctuations of the particles, the particles tend to first form the network structures (part b) rather than clusters (part c), because they are interacting each other in the electrostatic field where the stress built-up by the fluctuations is borne only by the particles. In the stress relaxation process, solvent-rich regions may be created via a local solvent-squeezing process, i.e., a process involving squeezing of solvents from the regions rich in the particles, i.e., the regions where the local concentration of the particles is higher than the average concentration, into the regions poor in the particles, resulting in the network structure formation. The network structure coarsens with time. In the long time (hydrodynamic) limit, the network shown in part (b) will be eventually transformed into clusters shown in part (c). This phenomenon may be treated as phase separation coupled by the viscoelastic relaxation of the colloidal particles.

A similar network formation was reported in phase-separation processes of polymer solutions. For example, poly(vinyl methyl ether) (PVME) in water, a highly dynamically asymmetric system, showed a network-type structure rich in polymers in the matrix of water as a consequence of stress–diffusion coupling and viscoelastic phase separation [16]. The network structures grow via the break-up processes of a part of the network or thread that bridges neighboring networks and the degeneration processes of the broken part of networks to the neighboring network. The break-up and degeneration processes cause coarsening of the network structure so as to reduce the total interface area and hence the interfacial free energy of the threads.

The break-up processes involve a built-up of local stress, and the degeneration processes involve the local stress relaxation. In the growth process, the entangled polymer chains in the threads are pulled against each other by *rubber-like elastic forces* and thinner parts of the thread will be broken. As a consequence, threads between the network junctions do not have either uniform diameters or smooth interfaces against the water phase, and the mesh size of the network has a broad distribution. Furthermore, growth of the network size with time does not exhibit dynamical self-similarity. These characteristics are believed to be a consequence of the threads being composed of entangled polymer solution where the entangled polymer chains are in the field of *rubber-like elastic forces* in the course of this pattern formation. This elastic force field extends over a large length scale through the network structures of threads.

Note that the patterns reported in the PVME/water system having a characteristic length of about 50 μm are already very large and macroscopic, reflecting a very late stage of phase separation. What happens in much earlier stages of the phase separation? What can be learned from a very early stage of phase separation in such systems? How different is the behavior from that for a dynamically symmetric system? These fundamental issues have not yet been well-explored and are not well-understood. Therefore, a series of work along this line [17, 18] has been conducted.

*Figure*

*8-9*represents time-resolved light scattering profiles in the course of phase separation via spinodal decomposition for (a) a semi-dilute solution of very high molecular weight PS in DOP [17], a typical system having the dynamical asymmetry, and (b) a binary polymer mixture of PB and SBR, poly(styrene-

*r*-butadiene), a dynamically symmetric system [19].

*I*(

*q*,

*t*) is the light scattering intensity distribution at time

*t*after phase separation started where

*q*is the magnitude of the scattering vector

**q**, defined by \(q = (4{\rm{\pi }}/\lambda )\sin (\theta /2)\) with

*λ*and

*θ*being the wavelength of incident light and the scattering angle (both in the sample), respectively. PS has

*M*

_{w}of 5.48 × 10

^{6}, and the PS concentration is 6 wt%, equal to 6.7 times overlap concentration [10]. Hereafter this system shall be designated as PS548/DOP 6.0 wt%.

*Figure*
*8-9a*
shows the result at quench depths of Δ*T* = 3.0°C below *T*
_{cl}. In the early stage, shortly after the quench (see the top part of (a)), a very broad scattering maximum appears; the maximum intensity increases with time with the peak position with respect to *q* (defined *q*
_{
m
}) almost unchanged, indicative of an early stage spinodal decomposition (SD) [20–22] where concentration fluctuations of a very large characteristic length of about 3 μm are developed for this system. As time elapses, the peak position *q*
_{m} rapidly shifts toward small *q* and peak intensity further increases (see the bottom part of (a)). It is quite striking to note that the low *q* modes increase their intensity but the high *q* modes do not. This time-evolution behavior is very different from that observed for dynamically symmetric mixtures of simple liquids or polymers, as will be described immediately below.

*Figure*
*8-9b*
shows typical results for nearly dynamically symmetric polymer mixtures. In the early stage SD (top part), particular Fourier modes of the concentration fluctuations are selected to grow so that the scattering peak becomes sharper with time without changing *q*
_{m}, in contrast to *the asymmetric system* where the peak tends to become broader with time. In the late stage (bottom part of *Figure*
*8-9b*
) mode-coupling occurs so that high *q* modes decay and low *q* modes grow; consequently, the peak position *q*
_{m} shifts toward low *q*. In contrast to this trend, for the asymmetric mixture only the low *q* modes grow, and as a consequence the peak becomes broad and shifts toward small *q*. Apparently the profiles at different times in the bottom part of *Figure*
*8-9a*
cannot be scaled with the time-dependent *q*
_{m} so that the domain-growth is not dynamically self-similar, albeit the time-evolution of the profiles in the bottom part of *Figure 8-9b* clearly scales with the time-dependent *q*
_{m} [21–23]. What do these unique characteristics mean? In order to understand the fundamental differences in the two systems, the growth rate of various Fourier modes in the early stage SD has been theoretically analyzed as a function of *q*.

*ξ*

_{0}, as shown in

*Figure*

*8-10(a)*where polymer molecules are schematically illustrated by lines, and solvent molecules are represented by small open circles or coarse-grained by a blue medium [17]. Consider the case where concentration fluctuations of polymer molecules with a characteristic length

*r*are built-up in the system by thermal excitation (see

*Figure*

*8-10b*), so that the system has regions slightly rich in polymers and regions slightly poor in polymers, represented respectively by the darker and brighter blue regions.

The concentration fluctuations will build up stress, and this stress is borne only by the polymers, since solvent molecules relax much faster than polymers. The built-up spatial variations of local stress will contribute to the free energy functional and hence to the cooperative diffusion of polymers, which in turn will affect the concentration fluctuations of polymers and hence the pattern formation of the system, suppressing the dynamics of the fluctuations, because building up of the concentration involves the extra cost of stretching free energy. This intuition about the suppression will be proven to be correct later in (8).

Consider the case where the concentration fluctuations of a large-length scale as shown in *Figure*
*8-10c*
are thermally excited so that there are entangled networks with mesh size *ξ* smaller than *ξ*
_{0} in polymer-rich regions and networks with mesh size *ξ* larger than *ξ*
_{0} in polymer-poor regions. Such fluctuations will be developed only after very large length-scale molecular rearrangements and will inevitably occur very slowly. Under this situation, the stress developed in the systems will be completely relaxed at the time when such concentration fluctuations are built-up via the long-range rearrangements of entangled polymers. Thus the stress–diffusion coupling is completely screened out in this case, leading to the concept of a “screening length” for the stress–diffusion coupling effects. This screening length was named the *viscoelastic length ξ*
_{ve} [12]. The stress–diffusion coupling is screened out if *r* > *ξ*
_{ve}, but if *r* < *ξ*
_{ve} the coupling is effective. This length should depend on dynamical properties of component elements, as shown quantitatively by (
*13*
) later.

### 3.3 Basic Time-Evolution Equation and a Theoretical Analysis of the Early Stage Self-Assembly in Dynamically Asymmetric Systems

*δϕ*(

*r, t*) ≡

*ϕ*(

*r*,

*t*) −

*ϕ*

_{0}represents fluctuations of local composition

*ϕ*(

*r*,

*t*) of a component (e.g., A) in mixtures at position

*r*and time

*t*away from its average value

*ϕ*

_{0}. Here it is assumed that these systems are isotropic for simplicity.

*F*=

*F*{

*δϕ*} is the free energy functional of

*δϕ*(

*r*,

*t*), and

*δF*/

*δϕ*is the variational derivative of

*F*with respect to

*δϕ*(

*r*,

*t*). α

_{a}is the so-called

*dynamical-asymmetry parameter*defined by

*D*

_{K},

*N*

_{K}, and

*ϕ*

_{K}are, respectively, self-diffusivity, DP, and volume fraction of the

*K*-th component in mixtures (

*K*= A or B). In (6), \(\alpha _{\rm{a}} = \phi _0 ^{ - 1}\) for polymer solutions, \(\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}}\over \sigma } \;(r,t)\) is the local stress tensor, and

*ζ*(

*r*,

*t*) is the random thermal noise term expressed by the following fluctuation-dissipation theorem:

*k*

_{B}and

*T*are the Boltzmann constant and absolute temperature, and <

*x*> denotes the thermal average of quantity

*x*. Cahn–Hilliard–Cook (CHC) [20, 26, 27] and Ginzburg–Landau(GL) [28] originally introduced the first and third terms on the right-hand side (rhs) of (5), while Kawasaki–Ohta introduced the hydrodynamic term [29]. Doi–Onuki further generalized the equation by incorporating the stress term associated with the dynamical asymmetry and the stress–diffusion coupling (the second term on the rhs of (5)) [24].

*D*

_{A}

*N*

_{A}=

*D*

_{B}

*N*

_{B}, and hence α

_{a}= 0, the second term on the rhs of (5) vanishes. Consequently the equation reduces to the well-known

*CHC equation or time-dependent GL (TDGL) equation*[30]. If

*δϕ*(

*r*,

*t*) is small and the hydrodynamic term can be neglected, (5) can be linearized, and the linearized equation in

*q*-space is given by

*r*

_{0}≡ −(∂

^{2}

*f*/∂

*ϕ*

^{2})

_{0}(with

*f*being the free energy density of the mixture) is a parameter related to the thermodynamic driving force for composition fluctuations. The driving force is positive in a phase-separation condition, which therefore tends to grow fluctuations, while it is negative for mixtures in a single-phase state, which hence tends to decay thermally activated fluctuations.

*C*

_{g}is a positive constant related to the gradient free energy associated with the nonlocality of interactions [10, 26].

*G*(

*t*) is a relaxation function for the shear modulus given by

*G*

_{i}and

*τ*

_{ i }are the strength and relaxation time for the i-th relaxation process.

*ζ*(

*q,t*) satisfies the fluctuation dissipation theorem in

*q*-space,

The second term on the rhs of (8) represents the contribution of the stress built-up by the increase of composition fluctuations \(\partial \delta \phi (q,t')/\partial t'\) to the time-evolution of concentration fluctuations. Since the integral is always positive, the stress term tends to suppress the growth rate of composition fluctuations in two-phase mixtures [17] or the relaxation rate of the fluctuations in single-phase mixtures [18], as is naturally anticipated. Thus interestingly enough, the dynamics is affected by the stress relaxation process and hence by the mechanical properties of the system.

*q*≪

*R*

_{g}where

*Λ*(

*q*) in (8) becomes independent of

*q*and equal to

*Λ*(0), a limiting value at

*q*= 0, the following is obtained:

*ξ*

_{ve}defined by

*viscoelastic length*.

*η*

_{0}is the zero-shear viscosity given by

*ν*

_{0}is defined by

*ν*

_{K}(

*K*= A or B) being the molar volume of

*K*-th monomeric unit.

The *viscoelastic effect* gives a *renormalization effect* on the Onsager kinetic coefficient *Λ*(0) via the term of \((1 + \xi _{{\rm{ve}}}^2 q^2 )^{ - 1}\), giving rise to an effective suppression of *Λ*(0). This suppression depends on a product of the viscoelastic length *ξ*
_{ve} and *q*; the larger the value is, the larger is the suppression. This explains the *anomalous time-evolution behavior* in the early stage of SD as already discussed in conjunction with the upper part of *Figure*
*8-9a*
in comparison with the normal behavior shown in the upper part of *Figure*
*8-9b*
. *ξ*
_{ve} for the system in *Figure*
*8-9a*
was found to be very large [17], as large as *ξ*
_{ve}/*R*
_{g} = 15.6. *ξ*
_{ve} is a unique and important characteristic length of the system which depends on the dynamical properties, most importantly on the dynamical-asymmetry parameter *α*
_{a}, but it also depends on the zero shear viscosity and the small *q*-limit of the Onsager coefficient.

It is important to note that the effects of dynamical asymmetry disappears in the case when *q ≪*1/*ξ*
_{ve}, or *r* ≫*ξ*
_{ve} or *t* ≫ \(\tau _{ve} \equiv \xi _{ve}^2 /D_{{\rm{app}}}\) (*τ*
_{ve} and *D*
_{app} being respectively the relaxation time for the viscoelastic effects, *viscoleastic time*, and cooperative diffusion coefficient). In other words, the dynamically asymmetric effects are insignificant in the hydrodynamic regime of a large space-time scale of observation.

*R*(

*q*) for

*q*-Fourier modes of the concentration fluctuations in the early stage SD process of the PS/DOP system shown earlier in

*Figure*

*8-9a*. From the data shown in the upper part of

*Figure*

*8-9a*,

*R*(

*q*) in (12) was evaluated as a function of

*q*at phase separation temperature

*T*= 10.8°C (or at a quench depth Δ

*T*=

*T*–

*T*

_{cl}= 3.0°C,

*T*

_{cl}being cloud point temperature), and the result was plotted in

*Figure*

*8-11a*(cross symbols) [17]. The figure also contains

*R*(

*q*) versus

*q*at

*T*= 12.3°C or (Δ

*T*= 1.5°C) (open circles). The solid lines in

*Figure*

*8-11a*show the theoretical predictions best-fitted with the experimental results using (12) with

*ξ*

_{ve}= 1.0 μm and 0.9 μm for the PS/DOP system at 10.8°C (the solid line fitted to cross symbols) and at 12.3°C (the solid line fitted to open symbols), respectively. The figure also represents the fictitious cases which ignored the viscoelastic suppression for

*R*(

*q*) by the broken line at 10.8°C and the dotted line at 12.3°C. It is striking to note a large suppression of

*R*(

*q*) at \(q\; > \;\xi _{{\rm{ve}}}^{{\rm{ - 1}}} \cong 1 \times 10^{ - 3} \;nm^{ - 1}\) compared with the case of

*ξ*

_{ve}= 0.

*ξ*

_{ve}is as large as 1.0 μm and about 15 times larger than

*R*

_{g}, \(\xi _{{\rm{ve}}} /R_{\rm{g}} \cong 15\). Owing to the large suppression of

*R*(

*q*) at

*qξ*

_{ve}> 1, the Cahn plot in

*Figure*

*8-11b*exhibits a strong downward deviation from the linear relationship between

*R*(

*q*)/

*q*

^{2}versus

*q*

^{2}expected for the fictitious cases of

*ξ*

_{ve}= 0 as shown by broken and dotted straight lines. This deviation is another indication of the strong suppression of

*R*(

*q*) at

*qξ*

_{ve}> 1.

### 3.4 General Background on the Effects of Shear Flow on Self-Assembly of Both Dynamically Symmetric and Asymmetric Systems

*R*(

*q*) inherent in the systems.

*R*(

*q*) generally depends on modes of concentration fluctuations or composition fluctuations and hence depends on

*q*as predicted by (12). In the single phase state of solutions and mixtures,

*r*

_{0}is negative and

*C*

_{g}is always positive so that

*R*(

*q*) becomes negative; hence, the thermally activated fluctuations decay and the decay rate \(\Gamma (q) \equiv - R(q)\) is quartic with respect to

*q*in the case of

*ξ*

_{ve}= 0 but quadratic with

*q*at

*qξ*

_{ve}≫ 1,

*q*values the faster the decay rate

*Γ*(

*q*) in any cases as depicted by the broken and solid curves in

*Figure*

*8-12*.

In *Figure*
*8-12*
, the shear-rate-dependent characteristic wave number *q*
_{c,shear} is defined as the *q* value satisfying \(\dot \gamma = \Gamma (q)\). This value *q*
_{c,shear} depends on *ξ*
_{ve}. If Fourier modes with their wave number *q* (*q*-modes) satisfies *q* > *q*
_{c,shear}, \(\Gamma (q) > \dot \gamma\) so that these fast modes of fluctuations disappear before shear deforms the fluctuations and hence are not affected by shear. However the *q*-modes with *q* < *q*
_{c,shear} satisfy \(\Gamma (q) < \dot \gamma\), so that these slow modes of fluctuations do not relax but rather stay under shear flow and hence are deformed and affected by shear.

*q*

_{cT}is the characteristic wave number of the system defined by

*q*

_{c,T}corresponds to the inverse of the thermal correlation length \(\xi _{\rm{T}}\). If

*q*

_{c,shear}<

*q*

_{c,T}, only Fourier modes of small wave number

*q*are affected by shear, thus the system as a whole is not affected by shear. In this case the shear is defined as “weak shear” for this system.

Unique points in polymers are anticipated to arise from *Γ* being very small compared with the values for small molecules. Hence even *very small shear rates affect polymer systems* compared with the case of small-molecule systems. In the dynamically asymmetric systems, the viscoelastic effects further suppresses *Γ* as schematically shown in the *Figure*
*8-12*
. Hence the systems become even more sensitive to small shear rates.

## 4 Methodology

This section describes some examples of shear-rheo-optical apparatuses which enable real-time and in situ exploration of dissipative structures developed in the systems under shear flow together with shear stress and normal stress.

### 4.1 Simultaneous Measurements of Stress, Optical Microscopy, Light Scattering, Transmittance, Birefringence, etc

We paid special attention to design, as much as possible, a versatile apparatus which permits *simultaneous measurements* of rheological properties, light scattering, and transmission optical microscopic images for a wide range of systems having very low shear and normal stresses to very large ones, very weak scattering intensity and optical contrast to very high scattering intensity and contrast. For this purpose one of the most reliable rheometers commercially available was modified and a scattering apparatus and an optical microscope were installed to the rheometer [31]. The apparatus takes in situ a two-dimensional (2D) light scattering pattern, an optical micrograph, and rheological properties under continuous shear, oscillatory shear, and step-up and step-down in shear rate.

Data acquisition of scattering patterns and micrographs can be synchronized with a given strain phase *ϕ*
_{s} under dynamic shear field, which enables strain-phase dissolved measurements. Here *ϕ*
_{s} is defined by \(\phi _s \equiv \sin ^{ - 1} (\gamma /\gamma _0 )\), where *γ* is a strain and *γ*
_{0} is a strain amplitude. Charge coupled device (CCD) cameras are used as detectors for light scattering and for optical microscopy measurements, which enable detection of quantitatively 2D scattering patterns with a weak intensity level and optical micrograph with a weak contrast level.

*Figure*

*8-13*shows an overall view of the apparatus. The base of the apparatus is a modified version of the ready-made rheometer RMS-800 (Rheometric Scientific Inc.) A in

*Figure*

*8-13*. The rheometer was modified in order to install a SALS (B) and a transmission light microscope (C).

*Figure*

*8-14*shows a close up of its main parts (B) and (C) in

*Figure*

*8-13*; (B) on the right-hand side of a shear cell (part b) and (C) on the left-hand side.

*Figure*

*8-15*shows the scheme of the whole apparatus. The rheometer, the shear SALS, and the shear microscopy were controlled by using exclusive computers A5, B18, and C14, respectively. Additionally, another computer D

_{1}was used for synchronization of the optical and scattering measurements with the rheological measurements. The flow chart of the data and that of the control signal are shown as black and grey arrows in the figure respectively.

*Rheometer*. A rheometer RMS-800 allows various kinds of rheological measurements, i.e., constant shear, oscillatory shear, constant stress, stress relaxation, and application of an arbitrary wave shape of strain. The position of the transducer was modified (part (a) in *Figure*
*8-14*
, part A3 in *Figure*
*8-15*
); the transducer was set at a position higher than that for a normal RMS-800 by 110 mm to expand a working space for installation of the optical and scattering instruments. Two types of shear cell (part (b) in *Figure*
*8-14*
, part A1 in *Figure*
*8-15*
) were made, a parallel plate, and a cone-and-plate, made of transparent quartz for optical measurements. Their diameter is 80 mm and the cone angle of the cone is 1.00°. The lower plate or cone is mounted on an actuator (A4) for an application of a strain. The upper plate fixture is mounted on the transducer (A3), which allows measuring both torque and normal force. The capacity of the transducer is from 2 × 10^{−5} to 0.2 Nm for a torque measurement and from 2 × 10^{−2} to 20 N for a normal force measurement, respectively. Therefore, it is possible to measure not only the viscosity of water ∼1 mPa but also polymer in bulk having a high shear modulus ∼10^{9} Pa. However, it is necessary to care about the machine compliance effect, which may be important for *G′* > 10^{4} Pa at *ω* > 0.1 rad/s with such a large diameter cone-and-plate.

*Shear-SALS*. The whole optic of the shear-SALS instrument (part B in *Figures*
*8-13*
and
*8-15*
) is mounted on a set of parallel rails laid on the right-hand side of the rheometer in order to facilitate an alignment of the optics on the rails: The optics can be slid in and out from the rheometer. As shown in *Figure*
*8-15*
, a 10 mW He-Ne laser (B1) is used as an incident light source (wavelength in vacuum *λ*
_{0} = 632.8 nm). A beam splitter (B2) is located to monitor the intensity of the incident beam by a photodiode (B3). The incident laser beam can be attenuated by a set of neutral density (ND) filters (B4) with an intensity range of 0.01–100% to avoid intense scattered beam coming into the detector. A *λ*
_{0}/4 plate (B5) and a polarizer (B6) are provided to rotate a polarization direction of the polarized incident beam. After passing through a pinhole (B7) for eliminating stray light, the incident beam is reflected upwards by a mirror (B8), passes through, and irradiates a sample between a cone-plate or a plate-plate fixture. A small mirror (B9) is placed behind the fixture in order to block the transmitted incident beam from entering into the detector (B16). Intensity of the transmitted beam reflected by the mirror is also used to measure by a photodiode (B10), which allows to measure in situ the turbidity of the sample. An analyzer (B11) is set to study polarized and depolarized light scattering intensity distributions.

Since an active surface area of a cooled CCD camera (B16) is small (13.82 × 13.82 mm^{2}), a focusing optical system is mounted in the instrument. This system consists of two lenses as a condenser for scattered light. The first aspherical len (B12) placed behind the analyzer and the second biconvex len (B14) collect and collimate the scattered light into the CCD camera B16. Because of a space limitation, the aspherical lens is cut near the middle and a large flat mirror (B13) is mounted between two lenses at a 45° angle to the incident beam. A neutral density filter (B15) is available for attenuation of the scattered light in the range of 0.01–100%.

Focusing a scattering pattern directly on the cooled CCD camera (B16) and a sufficiently large dynamic range of the detector (16-bit) enabled us to quantitatively detect scattered intensity distribution of a 2D scattering pattern. The cooled CCD camera has an active surface area of 13.82 × 13.82 mm^{2} with 512 × 512 pixels (27 × 27 μm^{2}/pixel). It is possible to set any rectangular areas on the CCD chip as a unit for data acquisition, which allows a rapid data acquisition and a reduction of the data size. Usually the area of 280 × 512 pixels is used because the half-size aspherical lens is used. It is also possible to divide an accessible CCD chip area by binding a rectangular data area composed of *xy* pixels as a unit for data acquisition to reduce data size, where *x* and *y* are any integers satisfying 1 ≤ *x*, *y* ≤ 512.

The minimum exposure time of the detector is 50 ms and the minimum acquisition time for one measurement consisting of 280 × 512 pixels is 2.0 s. Therefore a time resolution between measurements is about 2.0 s, which is sufficient for kinetic studies of phase separation behavior in polymer melts and solutions. It is possible to take data with a shorter time resolution by accessing a smaller CCD chip area, for example, accessing a one-dimensional (1D) area parallel or perpendicular to the flow direction, and/or by binding the pixels as described above.

A CCD camera controller (B17) is connected to a computer (B18), which allows various modes of measurements for both steady and dynamic measurements as follow; a measurement with a fixed time interval, a measurement with a triggered signal, and a measurement with a programmed timetable.

*Shear microscopy*. A transmission optical microscope (part C in *Figures*
*8-13*
and
*8-15*
) was designed specially for the rheometer. The whole optics is mounted on a set of parallel rails, laid on the left-hand side of the rheometer, just similarly to the shear-SALS instrument. A halogen lamp (C1) (12 V 100 W, illuminance of 30,000 lux) is used as a light source. The incident beam from the lamp passes through a flexible optical fiber (C3) and is reflected up into the sample by the mirror (C4). A heat-absorbing filter (C2) is set in the halogen lamp unit to avoid heating the sample with the intense incident light beam. Three kinds of long-focus type objective lenses (C6) (10×, 20×, and 50×) are available for the observation of submicron-size structures. A polarizer (C5) can be set above the mirror (C4) and an analyzer (C10) in the focusing microscope body (C8), which makes observations under various polarization conditions possible.

Because of a limitation of the working space available to the rheometer, a focusing microscope body (C8) is bent toward a CCD camera (C11). In the focusing microscope body, two mirrors (C7 and C9) are set at 45° against the objective lens to bend the light-beam pass. The CCD camera is connected to an analog image processing unit (C12). It should be noted that the CCD camera (C11) is not the same as the one (B16) used for the shear-SALS. C12 can perform real-time image processing for obtained micrographs, e.g., contrast enhancement and background subtraction, etc. Therefore, a high contrast micrograph can easily be taken even when structures in a sample themselves had a weak contrast due to a small difference in refractive indices between the structures of interest and their surrounding medium.

C12 is connected in parallel to a video tape recorder (VTR) (C13) for continuous analog recording and to a computer (C14) with a video board for digital image capturing with 640 × 480 pixels and 8-bit resolution. The public domain NIH image program enables not only automatic capturing of micrographs but also digital image analyses for these micrographs. The shortest data acquisition time per frame with 640 × 480 pixels is 1.0 s, which is short enough for observations of kinetic processes in polymeric systems under shear flow. Since objects under a continuous shear flow quickly move away from the camera window, an exposure time of the CCD camera should be short enough to take images. Therefore, the shutter speed of the CCD camera is set to 0.1 ms, during which time an object moves at most 5 μm in the case of a sample thickness of 0.5 mm and shear rate \(\dot \gamma\) of 100 s^{−1}. Image analyses and digital image processing are performed on thus saved micrographs by using NIH image and Adobe Photoshop 4.0J.

*Temperature enclosure*. A temperature enclosure (A2 in

*Figure*

*8-15*) was constructed with two windows: left-hand side for shear microscopy and right-hand side for shear-SALS to allow a simultaneous observation of both light scattering patterns and optical microscope images.

*Figure*

*8-16*shows a cross section of the temperature enclosure. Its design is basically the same as that used for previously developed shear-SALS apparatus [32, 33], except for new features associated with two windows and minor changes in dimensions. The enclosure has a double chamber structure in which temperature-controlled dry air circulates as shown by the arrows. Dry air heated by heaters enters into the outer and inner chambers. The flow of temperature-controlled air in the outer chamber shields the inner sample chamber from an ambient temperature change outside the enclosure. It also goes through shafts holding a cone-and-plate (or parallel plate) and then re-enters into the inner chamber in order to reduce a temperature gradient along radial direction of cone-and-plate. The temperature control of the air flow inside the outer chamber enables to more precisely control temperature of the cone-and-plate shear cell.

A cooling system is also available for measurements below room temperature. It cools dry air before entering the temperature enclosure. The experimental temperature range is from 0 to 200°C with an accuracy of ±0.3°C. Temperature variation along the radial direction was measured to be less than ±0.3°C. The uniformity of temperature over the shear cell was checked by a set of thermo-sensors embedded in the various place of the upper plate of the shear cell which is specially made only for testing the temperature uniformity. Such temperature accuracy is essential to investigate critical phenomena, which are much more sensitive to temperature than conventional rheological properties of systems which do not involve phase transitions.

*Transmittance*. Turbidity or transmittance of the incident beam for systems subjected to shear flow can be measured in situ by a photodiode (B10) as described already in section “Shear-SALS.” Birefringence can be also measured by detecting transmitted light intensity with the CCD camera (C11) under crossed polarizer (C5) and analyzer (C10) with their polarization directions oriented at ±45° with respect to the flow direction. A compensator should be inserted in between C11 and C10 in order to convert the transmitted intensity into birefringence.

It should be noted that the construction of the apparatuses become much simpler when one wants to measure only the shear-SALS and/or shear optical microscopy [32]. A one-dimensional photo-diode array may also be used for quantitative measurements of light scattered intensity distributions from systems under shear flow [34].

*Calibrations and corrections*. Various calibrations and corrections that need to be done for the shear-rheo-optical apparatus are described below (see [31] for further details).

The rheometer was calibrated according to the specification of the maker. The shear-SALS instrument with the cooled CCD camera was calibrated in terms of both scattering angle and scattering intensity. The scattering angle versus channel number of the CCD was calibrated by measuring diffraction patterns from two kinds of diffraction gratings (500 and 1,000 lines/inch). The range of the scattering angle *θ* detected by the CCD and the resolutions of *θ*, Δ*θ*, were determined to be 2.3°≤ *θ* ≤ 22.0° and Δ*θ* = 0.11° in arc, respectively. The scattered intensity measured with the CCD was calibrated for a variation of the solid angle subtended by one channel and uniformity of detector sensitivity. The calibration used fluorescence intensity having no *θ*-dependence which is emitted from an aqueous solution of ethylene blue with concentration of 10^{−5}mol/L when the solution is irradiated with a He-Ne laser (*λ*
_{0} = 632.8 nm). Through the calibration process, the effects of spherical aberration of the lens can be confirmed, if there were.

*Synchronization of light scattering measurements and optical microscope imaging with a shear-strain phase under oscillatory shear flow*. Under a dynamic shear mode, structures may change with strain phase *ϕ*
_{s}. In such a case, it is necessary to synchronize data acquisition for both scattering and microscopy with *ϕ*
_{s}. For this purpose, a computer is used as a synchronizer which takes an electronic signal of *ϕ*
_{s} from the control unit of the rheometer. The computer sends a triggering signal for data acquisition for both scattering and optical microscopy at a specified *ϕ*
_{s} and over a specified phase interval Δ*ϕ*
_{s}. The detailed calibration and correction methods required for the synchronized data acquisition should be referred to the original reference [31].

### 4.2 Examples: Simultaneous Measurements of Stress, Shear-SALS, and Shear-Microscopy

In most of the complex liquids as described here, the structures in the liquids and their rheological properties are generally very sensitive to thermal and shear histories given to the system. Therefore it is most desirable to conduct truly simultaneous measurements of the various properties and structures in situ for a given system in order to avoid complications brought by the effects that the measured various quantities reflect different states of the systems due to different thermal and/or shear histories. Examples of simultaneous measurements for the following two systems are described below.

*Continuous shear mode for the system exhibiting shear-induced single-phase formation*. The system to be discussed here was described earlier in

*Section*

*2.2*and showed disappearance of the strong scattering with increasing shear rate above the critical shear rate \(\dot \gamma\)

_{c,single}for single-phase formation in the same system.

*Figure*

*8-17*shows the \(\dot \gamma\) dependence of viscosity

*η*measured by the apparatus described in

*Section*

*4.1*. The figure also includes

*η*measured by ready-made rheometer ARES-FS (Rheometric Scientific Inc.) dedicated only to rheology experiments for comparison. The equality between two data within the error of ±5% can clearly be confirmed, indicating that the measurement of rheological properties works well with this rheo-optical apparatus. Thus this apparatus really allows measurement of such rheological properties as characterized by weak shear stress simultaneously with light scattering patterns and optical micrographs, both of which will be shown below (

*Figures*

*8-18*and

*8-19*, respectively).

*Figure*

*8-18*shows steady-state light scattering patterns simultaneously measured for the same solution as that in

*Figure*

*8-17*as a function of \(\dot \gamma\).

*Figure 2-19*shows the steady-state transmission optical micrographs measured simultaneously with the scattering patterns shown in

*Figure 8-18*and rheology shown in

*Figure*

*8-17*. It is important to confirm whether or not the obtained microscope images truly reflect the structures developed under the shear field. For this purpose, the 2D fast Fourier transformation (FFT) spectra were calculated from the images shown in

*Figure*

*8-19*, results of which are shown in

*Figure*

*8-20*. The \(\dot \gamma\) dependence of the FFT spectrum shown in

*Figure*

*8-20*is essentially consistent with that of the corresponding scattering patterns in

*Figure*

*8-18*. Hence, it can be concluded that the real-space images in

*Figure*

*8-19*captured by the microscopy truly reflect the phase-separated structures under shear flow.

*Figure*
*8-19*
shows that: large and nearly round droplets existing at the weak shear rate of \(\dot \gamma = {\rm{0}}{\rm{.0063}}\;{\rm{s}}^{ - {\rm{1}}}\) (part a) tend to become smaller droplets elongated and aligned parallel to the flow direction (e.g., part b at \(\dot \gamma \cong 0.025\;s^{ - 1}\)); they are percolated along the flow direction \((\dot \gamma = {\rm{0}}{\rm{.1}} - 0.25\;s^{ - 1} )\) (parts c and d) and eventually transformed into stable strings extended with almost macroscopic dimension along the flow direction \((\dot \gamma = 4.0 - 40\;s^{ - 1} )\) (parts e and f). The real-space images thus captured as a function of \(\dot \gamma\) is consistent with the shear-rate dependence of the SALS, as will be detailed later in *Section*
*5*
. Although not shown here when \(\dot \gamma\) is increased above \(\dot \gamma _{c,\sin gle}\), the streak-like scattering pattern along *q*
_{
z
}, as shown in *Figure*
*8-18f*
disappeared as shown in *Figure*
*8-5b*
, and the string-like domains in *Figure*
*8-19f*
also disappears. Since the interfaces of the domains align parallel to flow in the range of the shear rates covered by the viscosity measurements shown in *Figure*
*8-17*
, the interface does not contribute to the viscosity, giving rise to a weighted average of *η* for a homogeneous solution of PS/DOP and that of PB/DOP solutions with the same concentration as the blend solution.

*Oscillatory shear mode for the system exhibiting the shear-induced concentration fluctuations or phase-separation*. The system to be discussed here was also described earlier in *Section*
*2.2*
. The single phase solution of PS548/DOP 6.0 wt% exhibited a strong butterfly-type scattering pattern when \(\dot \gamma\) exceeds the critical shear rate \(\dot \gamma _{c,{\rm{x}}}\), as a consequence of shear-enhanced concentration fluctuations. Here the problem will be extended to the shear-enhanced concentration fluctuations under dynamic shear mode.

*Figure*

*8-21*shows a Lissajous curve, shear stress

*σ*against shear strain

*γ*, obtained with the rheo-optical apparatus described in

*Section*

*4.1*. The figure indicates a nonlinear rheological response. One of the reasons for the nonlinear behavior is, of course, a large strain amplitude

*γ*

_{ 0 }of 4.2. The large

*γ*

_{0}, which brings about shear-induced phase separation or concentration fluctuations, should contribute to the nonlinear response of

*σ*with

*γ*.

*Figure*

*8-22*shows the

*ϕ*

_{ s }dependence of the scattering pattern measured simultaneous with the results shown in

*Figure*

*8-21*. At a first glance, a clear butterfly pattern can be seen which shows characteristics of shear-induced phase separation or concentration fluctuations. Thus, a dynamic shear flow creates the shear-induced structures as in the case of the steady shear flow. Also, it is recognized that intense scattering appears in harmony with

*ϕ*

_{ s }. An integrated scattered intensity along the

*q*

_{ x }axis, \({\mathcal {i}}\)(

*q*

_{ x }) is calculated, to quantitatively analyze the

*ϕ*

_{ s }dependence of scattering intensity, where \({\mathcal {i}}\)(

*q*

_{ x }) is defined by

*I*(

*q*

_{ x },

*q*

_{ y }= 0,

*q*

_{ z }= 0) is scattered intensity along the

*x-*axis,

*q*

_{1}= 0.47 μm

^{−1}, and

*q*

_{2}= 3.36 μm

^{−1}.

*Figure*

*8-23*shows \({\mathcal {i}}\)(

*q*

_{ x }) as a function of

*ϕ*

_{ s }, indicating clearly a nonlinear response with

*ϕ*

_{ s }. As

*ϕ*

_{ s }increases, \({\mathcal {i}}\)(

*q*

_{ x }) gradually increases in the region from

*ϕ*

_{ s }= 0 to 9π/16 and decreases more quickly in the region from

*ϕ*

_{ s }= 9π/16 to 15π/16. This indicates that the concentration fluctuations induced by shear flow grow with an increase of \(|\gamma |\), and they decay quickly with the decrease of \(|\gamma |\), although a small phase lag exists. The quick decrease of the scattered intensity indicates a quick elastic recoil of the stretched entangled networks and a quick solvent back flows (counter-squeezing effect of solvents) driven by osmotic pressure as will be clarified later in

*Section*

*6*. Thus dynamic shear flow induces the oscillation of the concentration fluctuations in harmony with

*ϕ*

_{ s }. It illuminates an intriguing physical problem on how the nonlinearity in

*σ*is related to the nonlinearity in the scattering response. This can be a future problem.

*Remarks*. Up to this point in this section, the in situ observation of scattering patterns and microscope images in the *x-z* plane both under continuous shear mode and under oscillatory shear mode have been described. It is worth pointing out an apparatus which enables in situ observation of scattering in the *x-y* plane under the oscillatory shear mode. The apparatus shown below uses a shear-sandwich mode for small-angle X-ray scattering (SAXS) [35]. The sample surface can be rotated by 90° around *x-*axis for observation of the SAXS pattern in the *x-z* plane.

*Figure*

*8-24*shows a conceptual layout of the dynamic SAXS (DSAXS) system. The DSAXS system comprises a stage-mounted imaging-plate system and a hydraulic sample-deformation device. Details concerning the sample-deformation device have been given elsewhere [36].

A sample can be uniaxially elongated or subjected to shear deformation. *Figure*
*8-24*
shows the latter case. In either case the displacement of the actuator determines the macroscopic strain on the sample. The maximum load and maximum displacement are 490 N and ±7.5mm, respectively, and the operating linear frequency is between 0 and 60 Hz. The load on the sample is detected by a load cell, and the displacement of the actuator is measured using a linear variable differential transformer (LVDT). The temperature of the sample can be controlled at a constant value between 173 and 573 K by using a sample chamber made of a copper block with heating elements and a channel of liquid nitrogen.

*xy*stage which is driven by two stepping motors through an intelligent stepping-motor controller. The

*xy*stage is made of two sets of conventional translation stages which are placed orthogonally to each other. There is a 100 × 100 mm

^{2}square aperture for the scattered X-rays in front of the imaging-plate stage. An imaging plate (400 × 200 mm

^{2}) divided into eight sections (100 × 100 mm

^{2}) is used for detecting SAXS patterns. Each section is moved to the position of the aperture in turn, to record the SAXS pattern as shown in

*Figure*

*8-25*.

The time required for the translational movement of the imaging-plate stage along the *x-*axis or the *y* axis is ca. 0.5 s/100 mm. Therefore, time-resolved measurements up to eight time slices are possible in a time scale shorter than 1 s per frame. The total number of frames for two-dimensional SAXS patterns per one imaging plate can be increased, and the dead time between successive measurements can be shortened accordingly by using a smaller aperture and a shorter sample-to-detector length at the expense of the spatial resolution of the X-ray scattering patterns. The details of the controlled system, including SAXS data acquisition with the strain phase by using a personal computer, should be referred to the original paper [36].

*Figure*
*8-25*
demonstrates an example of the synchrotron dynamic SAXS method (SR-DSAXS) applied to a polystyrene-*block*-poly(ethylene-*alt*-propylene) copolymer having number-average molecular weight (*M*
_{
n
}) of 3.4 × 10^{4}, heterogeneity index (*M*
_{
w
}
*/M*
_{
n
}) of 1.3, and PS volume fraction in the copolymer of 0.103. The copolymer has spherical microdomains comprising PS block chains in a matrix of poly(ethylene-*alt*-propylene) (PEP) block chains. The amplitude of the dynamic shear strain is 50%, the static strain is 0%, and the angular frequency is 0.0944 rad s^{−1}. Measurements were carried out at room temperature where the PS spheres are in a glassy state.

The SAXS profiles before shear deformation show that the PS spherical microdomains of average radius *R* = 7.1 nm are in body-centered cubic lattice (bcc) with the Bragg spacing of 21.2 nm and with the *g*-factor characterizing the paracrystalline lattice distortion of 0.09 ≤ *g* ≤ 0.13 [37, 38].

Upon imposing an oscillatory shear deformation, the (110) and \((\overline 1 10)\) planes become oriented parallel and perpendicular to the *x-z* plane, respectively, resulting in a four-point SAXS pattern. The degree of orientation increases with the strain cycle. *Figure*
*8-25a*
shows the SAXS patterns at four strain phases *ϕ*
_{
s
} of 0, *π*/2, *π*, and 3*π*/2. These SAXS patterns were taken with an exposure time of 1 s for X-rays and at 110th strain cycle after imposing the oscillatory strain, at which the degree of (110) orientation reached a steady value. *Figure*
*8-25b*
shows a model used to describe the response of the lattice deformation at each strain phase and to explain the four diffraction spots appearing in the two-dimensional SAXS patterns. More details should be referred to the original paper [38].

## 5 Shear-Induced Mixing

Macroscopically phase-separated two-phase systems comprised of dynamically symmetric two components are next considered. The systems to be considered in this section have upper critical solution temperature (UCST) phase diagrams. When shear flow is applied to the systems under the condition where \(\dot \gamma < \Gamma _{{\rm{orient}}}\), the relaxation rate of molecular orientation, the systems would not exhibit either molecular orientation or stress due to the orientation. Even in this case, if this \(\dot \gamma\) is larger than the relaxation rate of the interfacial deformation, the interface is deformed and broken, giving rise to smaller domains with a larger interfacial area. Overburst small domains tend to grow, driven by the thermodynamic force (or interfacial tension), but overgrown large domains tend to burst, driven by the viscous force. The burst and growth and its repetition control the steady-state domain size and size distribution, as will be described later in this section.

If \(\dot \gamma\) can be further increased above *R*(*q*
_{m}), the growth rate of the dominant Fourier mode of concentration fluctuations at early stage of spinodal decomposition, the domain size becomes smaller than the critical size comparable to the interface thickness. Under this condition shear-induced mixing may take place. It should be noted that a considerable degree of intermixing of the two components in the mixtures may occur in each domain prior to the shear-induced mixing. In order for the shear-induced single-phase formation to occur, *R*(*q*
_{m}) should be sufficiently small so that the applied \(\dot \gamma\) can exceed *R*(*q*
_{m}) without inducing significant molecular orientation. The dynamically symmetric high-molecular weight mixtures having a small interfacial tension can fulfill this criterion. Under this condition \(({R} (q_{\rm m}) < \dot\gamma < \Gamma _{{\rm orient}})\), orientation and stresses developed in the two coexisting domains are equally divided and are relaxed at equal rates which are faster than the relaxation rate of the domain deformation. Hence the shear flow effectively affects the domain structures without causing a significant molecular orientation, i.e., under relaxed molecular orientations. The following sections will further elaborate this general picture as described above.

### 5.1 Shear-Rate Dependence of Steady-State Structures

The system to be discussed here is the one designated PS/PB(50/50)/DOP 3.3 wt% in *Section*
*2.2*
. DOP is a neutral solvent for PS and PB. DOP weakens repulsive segmental interactions between PS and PB. The ternary system can be approximately treated as a *pseudo-binary system* [39, 40] where a phase separation between PS and PB occurs in a medium of DOP and the thermally averaged local volume fraction of DOP is constant everywhere in space. In other words, a phase separation between polymers and solvent is insignificant. The pseudo-binary system is regarded to be equivalent to bulk systems when the segments of polymers in bulk are replaced by the *blobs* [10]. The pseudo-binary systems of polymers A and B are equivalent to bulk binary systems of A and B having effective DP of (*N*
_{A})_{b} and (*N*
_{B})_{b}, respectively. Here (*N*
_{A})_{b} and (*N*
_{B})_{b} are number of blobs for A and B polymers in the solution, respectively, and (*N*
_{A})_{b} = *N*
_{A}/*g*
_{s} and (*N*
_{B})_{b}/*g*
_{s} where *g*
_{s} is number of segments in the blobs.

The system has the following characteristics [40, 41]: *T*
_{cl} = 76°C (cloud point), the characteristic wave number *q*
_{m}(0) = 4 × 10^{−3} nm^{−1} or the characteristic wavelength \(\Lambda _{\rm{m}} (0) = 2{\rm{\pi }}/q_{\rm{m}} (0) = 1.5\;{\rm \mu} {\rm{m}}\), which characterizes the Fourier modes of the fluctuations with a maximum growth rate in the early stage of SD, collective diffusivity D_{app} ≅ 10^{−10}cm^{2}/s, characteristic time \(\tau \equiv [q_{\rm{m}}^2 (0)D_{{\rm{app}}} ]^{ - 1} = 6\;{\rm{s}}\), the characteristic interface thickness *t*
_{I} ≅ 100 nm, and interfacial tension between PS solution and PB solution *γ*
_{int} ≅ 10^{−7} N/m. The large *t*
_{I} value and the small *γ*
_{int} values are indicative of the system being in a very weak segregation limit. Thus the system can attain various states ranging from a macroscopically phase-separated state to a single-phase state under an experimentally accessible range of shear rates (0.005–500 s^{−1}). It has a viscosity of 0.63 poise (0.063 PaS), independent of shear rates from 0.2 to 400 s^{−1}.

*Figure*

*8-26*represents typical steady-state transmission optical microscope (OM) images and SALS patterns (shown in the insets) as a function of shear rate [41], while

*Figure*

*8-27*schematically represents a summary of the dissipative structures elucidated from a series of the studies as a function of shear rates [11, 32]. The results shown in

*Figure*

*8-26*were obtained by applying shear flow to macroscopically phase-separated solution of PS and PB in DOP across macroscopic interface parallel to the

*x*-

*z*plane as schematically shown in the left edge in

*Figure*

*8-27*(Regime I) and with an incident laser beam along the

*y*-axis (see

*Figure*

*8-3*for schematics of the experimental set-up and the Cartesian coordinate system). The patterns (a) to (f) were observed at steady state under a stepwise increase of shear rate. Snap shots of the OM images were taken at a shutter speed of 10

^{−4}s and a focal depth of about 10 μm, while the LS patterns were taken with an exposure time of 1/60 s.

At very small shear rates, smaller than 0.0063 s^{−1}, the macroscopic interface is conserved parallel to the *x-z* plane (the plane of the paper) as shown schematically in Regime I in *Figure*
*8-27*
, though the shear flow induces local perturbations of the interface [42]. The shear-rate range where the macroscopic interface is conserved corresponds to Regime I in *Figure*
*8-27*
. Upon increasing shear rate the macroscopic interface is broken and large nearly spherical droplets as shown in *Figures*
*8-26a*
and
*8-27a*
are developed as a consequence of a balance of viscous drag and interfacial tension. Corresponding SALS patterns appear at small angles and have an intensity distribution nearly independent of azimuthal angle*μ*. This shear-rate range corresponds to Regime II \((\dot \gamma < 0.0063\;{\rm s}^{ - 1} )\) in *Figure*
*8-27a*
.

At higher shear rates, droplets are broken into an increased number of smaller droplets elongated parallel to the flow direction so that the interfacial area is increased as schematically shown in *Figure*
*8-27b*
. The increased interfacial area is stabilized by the mechanical energy steadily imposed on the system. With increasing \(\dot \gamma\) the droplets become smaller and smaller and tend to align parallel to flow (*Figure*
*8-27b*
). The droplets eventually interconnected along the flow direction to result in formation of the percolated structure parallel to the flow direction as shown in *Figures*
*8-26b*
and
*8-27c*
. This is the so-called *shear-induced cluster-to-percolation transition phenomenon* [2] which develops an oriented percolated structure composed of solution rich in one polymer component in the matrix solution rich in other polymer component. The percolated structure is still unstable exhibiting a number of random irregularities as a consequence of such dynamical processes as frequent interconnections, disconnections, branching, and exchanges of domains through interactions with neighboring domains, as schematically shown in *Figure*
*8-27c*
. The scattering pattern is accordingly elongated normal to flow direction as shown in the inset of *Figure*
*8-26b*
. This shear-rate range is defined as Regime III \((0.063\; < \;\dot \gamma \; < 0.25\;s^{ - 1} )\) (*Figure*
*8-27b*
and
*c*
).

*string*structure [41, 43] with a macroscopic continuity of order of 500 μm along its axis parallel to the flow direction as shown in the OM image in

*Figures*

*8-26c*and

*8-27d*. The corresponding LS pattern is a sharp streak oriented normal to the flow direction as shown in the inset of

*Figure*

*8-26c*. The average diameter of the string decreases with \(\dot \gamma\) as may be seen in

*Figures*

*8-26c*and

*d*according to [44]

*τ*is the characteristic time of the system as defined earlier (6 s). The details will be described later in

*Section*

*5.4*. Upon increasing \(\dot \gamma\), the composition of the string and matrix phase deviate from the equilibrium one at \(\dot \gamma = 0\) in such a way that the intermixing of PS and PB increases so that the composition difference between the two phases decreases as manifested by the fact that the streak-like SALS pattern looses its intensity as shown in the insets of parts (c) and (d) in

*Figure*

*8-26*. These trends of the decreasing diameter and increasing intermixing of the strings are characteristic of Regime IV, shown schematically in the change from

*Figures*

*8-27d*and

*e*. At the critical shear rate, \(\dot \gamma _{{\rm c}, {\rm single}}\), where the shear-induced phase mixing occurs, \((\xi _ \bot )_d \simeq t_{\rm{I}} = O\;(100\;nm)\) for this particular system. At \(\dot \gamma > \dot \gamma _{{\rm c},{\rm single}}\), the system is brought into the single phase as evidenced by disappearance of the strings and streak-like LS patterns, though not shown in

*Figure*

*8-26*, but shown already in

*Figure*

*8-5b*and schematically shown in

*Figure*

*8-26f*. The details will be further discussed in

*Section*

*5.4*. This shear-rate range is defined as Regime V.

Similar changes in the dissipative structures with \(\dot \gamma\) were also reported later by Han and his coworkers. They found the change from droplets to string-like structure for a ternary system of PS/PB/DOP [45] and a binary system for low-vinyl PB/low-vinyl Polyisoprene (PI) mixture [46]. Furthermore by using low molecular weight polymer mixtures of PB/PI they found the string-like structure eventually goes into a homogeneous phase [47, 48].

### 5.2 Uniformity of Droplet Size in Regime II

*γ*

_{int}

*R*where γ

_{int}and

*R*are interfacial tension and radius of the droplet). However overgrown droplets will be broken into smaller ones due to the viscous drag (proportional to \(\dot \gamma \eta R^2\) where

*η*is the viscosity of droplets and matrix which are assumed to be identical and to be characterized by Newtonian fluids for simplicity). The droplets will repeat the processes of coalescence-growth and breakup before reaching steady-state average size and size distribution. A balance of the two opposing forces described above predicts the average size

*R*,

*Figure*

*8-28*shows light scattering intensity distributions

*I*(

*q*

_{ x }) versus

*q*

_{ x }(triangles) and

*I*(

*q*

_{ z }) versus

*q*

_{ z }(open circles) in the double logarithmic scale at the steady-state and at two representative shear rates [49]. The data were obtained by the rheo-optical apparatus described in

*Figures*

*8-13*

*–*

*8-16*. It is striking to note that

*I*(

*q*

_{ z }) at \(\dot \gamma = 0.143\;{\rm s}^{ - 1}\) oscillates with

*q*

_{ z }, showing at least 13 scattering maxima and minima over the

*q*range covered. Note that part (b) is an enlargement of a part of the profile in part (a) at a large

*q*

_{ x }and

*q*

_{ z }range which is encompassed by the square. On the other hand

*I*(

*q*

_{ x }) does not show such a remarkable oscillation as that in

*I*(

*q*

_{ z }), though there are some oscillation. The oscillation in

*I*(

*q*

_{ z }) sifts toward higher

*q*values at a higher shear rate as shown in part (c) obtained at \(\dot \gamma = 0.466\;{\rm s}^{ - 1}\). The oscillation in

*I*(

*q*

_{ x }) at this shear rate is more distinct than that at the lower shear rate; one can recognize three maxima or shoulders along

*q*

_{ x }

_{.}

*R*

_{ z }= 12.0 μm (a and b) and 3.9 μm (c) and with its axis of revolution oriented parallel to the flow direction. The best-fit between calculated

*I*(

*q*

_{ z }) for the spheroids having size distribution with respect to

*R*

_{ z }given by

*I*(

*q*

_{ z }) gives <

*R*

_{ z }> = 11.95 μm and \(\sigma _z / < R_z > \; = 0.025\) at \(\dot \gamma = 0.143\;{\rm s}^{ - 1}\) and <

*R*

_{ z }> = 3.95 μm and \(\sigma _z / < R_z > \; = 0.046\) at \(\dot \gamma = 0.446\;{\rm s}^{ - 1}\) [49], though the best-fitted calculated profiles are not shown in

*Figure*

*8-28*. It is quite striking that the size distribution becomes very narrow under shear flow. Compared with

*R*

_{ z }, the size distribution of

*R*

_{ x }is much broader. In Regime II, the aspect ratio <

*R*

_{ x }>/<

*R*

_{ z }> was found to be about 2, independent of \(\dot \gamma\), and <

*R*

_{ z }> was found to decrease according to the Taylor’s law, in proportion to \(\dot \gamma ^{ - 1}\) [49]. It should be noted that both

*I*(

*q*

_{ z }) and

*I*(

*q*

_{ x }

*)*have the Porod’s asymptotic behavior,

*t*

_{I}/<

*R*

_{ z }> =

*t*

_{I}/<

*R*

_{ x }> ≅ 0. It is interesting to note that the standard deviation,

*σ*

_{z}, of

*R*

_{ z }from its average value <

*R*

_{ z }> reaches a steady value at an extremely long time after stepping up to a given shear rate as shown typically in

*Figure*

*8-29*.

### 5.3 String Structure in Regime IV

*η*′ in the matrix of another Newtonian fluid with viscosity

*η*. The two fluids are assumed to be immiscible. The Tomotika’s linear stability analysis predicts growth of the amplitude of undulation,

*ξ*

_{un}, of the thread with time given by

*α*

_{max}is given by

*a*is diameter of the thread and

*G*is a universal function of the viscosity ratio

*η*′/

*η*.

The system discussed here has a very weak interfacial tension. The droplets developed in Regime II are elongated and aligned parallel to flow and eventually interconnected via the shear-induced cluster-to-percolation transition as discussed earlier in *Section*
*5.1*
. The percolated domains are unstable when \(\dot \gamma < \alpha _{\max }\) given by (24b), showing the random irregularity as discussed earlier. However, when \(\dot \gamma > \alpha _{\max }\), the instability is suppressed by shear to result in the stable string [41, 43, 45].

*a*), giving rise to undulating string with a wavelength

*λ*

_{string}and eventually splitting into a series of droplets aligned along the flow directions.

*Figure*

*8-30*shows typical time evolution of LS patterns and OM images taken in situ and at real-time after the cessation of shear [51, 52]. As shown in the OM images, the stable strings under the steady flow in part (a) are split into a series of spheres aligned along the original flow direction in the form of “pearl-necklace” as shown in part (c). The droplets diffuse and coalesce into larger droplets and a memory of the spatial alignment of the droplets along the flow gradually fades away as may be seen in part (e).

Correspondingly, a diffuse scattering appears above and below the sharp streak as shown in part (d). The diffuse scattering has a maximum intensity at *q*
_{mx
} along the original flow direction, reflecting inter-droplets spacing originating from *λ*
_{string}. The sharp streak is still conserved as a memory of the original string structure and due to the pearl-necklace alignment of the droplets. As time elapses an average size of the droplets increases, giving rise to a low *q* shift of the position *q*
_{mx
} of the diffuse scattering. Simultaneously the streak loses its intensity as the pearl-necklace alignment fades away. It is quite intriguing to note the following two points: (1) the diffuse scattering is analogous to the butterfly-type scattering pattern; (2) a *combined butterfly scattering pattern and streak-like scattering pattern* is analogous to the scattering pattern due to the *string-like structure* developed for shear-induced demixing systems, both of which will be extensively discussed in *Section*
*6*
for shear-induced phase-separation.

Transformation of the string to the pearl-necklace pattern after the cessation of shear from the shear-rate regime where the stable string is formed was extensibly explored and analyzed later by C.C. Han and his coworkers [53] and J. Mewis and his coworkers [54–60] for polymer mixtures.

### 5.4 Shear-Induced Phase Transition

The shear-induced phase transition was determined from both the two-phase state (in Regime IV) [40, 61] and the single-phase state (in Regime V) [40] by analyzing a series of LS profiles *I*(*q*
_{
z
}) along the neutral axis (*z*-axis) measured at steady-state as a function of shear rates. Han and his coworkers also reported the shear-induced mixing of polymer mixtures based on their studies of the effects of shear flow not only on single-phase mixtures in terms of thermal concentration fluctuations [62–65] but also on two-phase mixtures having low molecular weights [47, 48]. Higgins and coworkers also reported shear-induced mixing of polymer mixtures from the shear effects on two-phase states [66, 67].

A qualitative observation of the disappearance (appearance, respectively) of the streak-like SALS pattern with increasing (decreasing, respectively) \(\dot \gamma\) in regime IV (V, respectively) is insufficient, because the disappearance and appearance may possibly occurs when the scattering objects just comes out and in of the windows of the SALS observations, respectively. One must indicate quantitatively a change in scattering functions between the two-phase state in Regime IV and the single phase state in Regime V in order to unequivocally assess the critical shear rate, \(\dot\gamma\)
_{c,single}, for formation of the single phase.

*I*(

*q*

_{ z }) were characterized by squared Lorentzian functions (SQL) [44a]

*ξ*

_{⊥})

_{d}is the correlation length of domains along the

*q*

_{ z }axis which depends on the lateral size of the strings; the greater the size, the larger the value of (

*ξ*

_{⊥})

_{d}. The squared Lorentzian function suggests that the system has a random two-phase structure [68, 69] along

*q*

_{ z }over the

*q*

_{ z }range covered in the experiments. Obviously \(I(q_z = 0) \sim (\xi _ \bot )_{\rm{d}}^3\), and (

*ξ*

_{⊥})

_{d}depends on \({\rm{ }}\dot \gamma\). Hence the plots log

*I*(

*q*

_{ z }) versus log

*q*

_{ z }give a series of different curves when \(\dot \gamma\) is varied. However when

*I*(

*q*

_{ z }) and

*q*

_{ z }are reduced by shear rate-dependent

*I*(

*q*

_{ z }= 0) and \((\xi _ \bot )_{\rm{d}}^{ - {\rm{1}}}\), respectively, a shear rate-independent master curve was obtained [44a]

*ξ*

_{⊥}is (

*ξ*

_{⊥})

_{d}for the two-phase state and (

*ξ*

_{⊥})

_{fl}for the single-phase state (to be discussed later), respectively.

*Figure*

*8-31*shows the master curve of log

*F*(

*Q*

_{s}) versus log

*Q*

_{s}obtained at a quench depth of Δ

*T*(0) = 8 K for a quiescent solution and at various shear rates from 1.0 to 160 s

^{−1}(see the master curve labeled Regime IV) [44b].

*I*(

*q*

_{ z }) were characterized [40, 44b] by Lorentzian functions (

*Ornstein–Zernicke equation, OZ*) [70]

*I*(0)

_{fl}is the

*I*(

*q*

_{ z }= 0) in the single phase state, and (

*ξ*

_{⊥})

_{fl}is the thermal correlation length along

*q*

_{ z }for the thermal concentration fluctuations under shear.

*I*(

*q*

_{ z }= 0) and (

*ξ*

_{⊥})

_{fl}depends on shear rate: (

*ξ*

_{⊥})

_{fl}increases with decreasing \(\dot \gamma\) toward the critical shear rate \(\dot \gamma _{{\rm c},{\rm single}}\) for the shear-induced phase transition. However when

*I*(

*q*

_{ z }) and

*q*

_{ z }are scaled with

*I*(0)

_{fl}and \((\xi _ \bot )_{fl}^{ - 1}\) as given by (27), a shear rate-independent master curve was obtained [44b]

*Figure*
*8-31*
also represents the master curve obtained for the results at Δ*T*(0) = 6 K and at a range of shear rates from 80.0 to 147 s^{−1} as well as that at Δ*T*(0) = 8 K and at 253 s^{−1}.

Thus it is revealed that the shear-induced phase transition dramatically changes the nature of the scattering function *I*(*q*
_{
z
}) along the neutral axis. This change in the scattering functions together with the shear-rate dependence of (*ξ*
_{⊥})_{d}, (*ξ*
_{⊥})_{fl}, *I*(0)_{d}, and *I*(0)_{fl}, as will be discussed immediately below, enable unequivocal determination of the shear-induced phase transition. For the fixed *q*-range of observation with the shear rheo-optical apparatus described in *Section*
*4.1*
, the observed profile *I*(*q*
_{
z
}) undergoes a unique shift over the master curve with increasing \(\dot \gamma\) from regime IV to V as follows: The profile *I*(*q*
_{
z
}) starts from the tail part of the SQL master curve at large *Q*
_{s} toward the head part of the SQL at small *Q*
_{s}, because (*ξ*
_{⊥})_{d} decreases with increasing \(\dot \gamma\); upon further increase of \(\dot \gamma\), the phase transition occurs and the profile discontinuously shifts to the tail part of the OZ mater curve at large *Q*
_{s}, which is followed by shifting toward the head part of the OZ at small *Q*
_{s}, because (*ξ*
_{⊥})_{fl} again decreases with increasing \(\dot \gamma\).

*Figure*

*8-32*shows the shear-rate dependence of

*ξ*

_{⊥}[(

*ξ*

_{⊥})

_{d}and (

*ξ*

_{⊥})

_{fl}] (part a) and

*I*(

*q*

_{ z }= 0) [

*I*(0)

_{d}, and

*I*(0)

_{fl}] (part b) [44b]. With increasing shear rate in the two-phase state (the data points showed by filled circles), the power law exponent

*n*in (20) appears to change from 1/4 to1/2 and eventually to 1/3 (in regime IV). Above the critical shear rate of \(\dot \gamma _{{\rm c},{\rm single}} = 127\;{\rm s}^{ - 1}\) (\(\dot \gamma _{\rm c}\) in the figure) for the shear-induced phase transition,

*ξ*

_{⊥}= (

*ξ*

_{⊥})

_{fl}increases discontinuously, which is followed by a monotonic decrease with increasing \(\dot \gamma\) in the single phase region, as shown by the open circles in

*Figure*

*8-32a*. As shown in part (b),

*I*(0)

_{d}decreases with \(\dot \gamma\) as shown by solid circles in the two-phase state, and

*I*(0)

_{fl}also decreases as shown by open circles.

*T*

_{c}(0) being the transition temperature at \(\dot \gamma = 0\) as shown in

*Figure*

*8-33*below. If the shear flow drops the phase transition temperature from

*T*

_{c}(0) to \(T_{\rm{c}} (\dot \gamma )\), (

*ξ*

_{⊥})

_{fl}, and

*I*(0)

_{fl}under the shear flow can be predicted by replacing Δ

*T*(0) with \({\rm{{\rm \Delta} }}T{\rm{(}}\dot \gamma {\rm{)}}\) given by

*Figure*

*8-33*for the definition), so that instead of (30) one should use

*Section*

*5.5*, where the proportionality constant has the dimension of square root of time.

*Figure*

*8-33*one should note that \({\rm \Delta} T_{\rm{c}} (\dot \gamma ) = {\rm \Delta} T(0)\) at \(\dot \gamma = \dot \gamma _{{\rm{c,single}}}\). The \(\dot \gamma _{{\rm{c,single}}}\) is re-defined here as \(\dot \gamma _{\rm{c}}\) for the sake of convenience, so that this fact together with (33) give

*Figure*

*8-32b*for

*I*(0)

_{fl}represents the best-fit of (36) with the experimental data

*I*(0)

_{fl}.

*I*(0)

_{fl}tends to increase and diverge as \(\dot \gamma\) is decreased to \(\dot \gamma _{\rm{c}}\), which is shown by the vertical dotted line. The solid line in

*Figure*

*8-32a*for (

*ξ*

_{⊥})

_{fl}represents the best-fit of (36) with the experimental data (

*ξ*

_{⊥})

_{fl}. (

*ξ*

_{⊥})

_{fl}also tends to increase and diverges as \(\dot \gamma\) is decreased to \(\dot \gamma _{\rm{c}}\).

*I*(0)

_{d}in regime IV should be given by [44b]

*I*(0)_{d} can be predicted by using (35), (37), and the experimental result on (*ξ*
_{⊥})_{d} shown in *Figure*
*8-32a*
. The predicted result shown by the solid line labeled by *I*(0)_{d} is consistent with the experimental result (see *Figure*
*8-32b*
), except for the lower shear-rate region which belongs to Regime III. Thus the \(\dot \gamma\) dependence of *I*(0)_{fl} and (*ξ*
_{⊥})_{fl} in Regime V as well as that of *I*(0)_{d} in Regime IV can be consistently interpreted in terms of the apparent downward shifting of the phase boundary with \(\dot \gamma\) as show in *Figure*
*8-33*
.

It may be worth-noting here that the following intriguing phenomenon has been reported with respect to the shear-induced phase transition: given polymer mixtures exhibit not only the shear-induced mixing but also shear-induced demixing. This is observed for mixtures of PS and PVME reported first by Winter and coworkers [71, 72]. Higgins and coworkers [73–75]. Chopra et al. [76] also reported both shear-induced mixing and demixing for given polymer mixtures.

*SD under shear flow*[44b, 77]. For example, after a shear-rate drop from Regime V (shear induced single-phase) to Regime IV, it is possible to

*extend*the following

*scaling hypothesis in the quiescent state to that under shear flow*. In the quiescent state, the coarsening of the phase-separated domains via SD can be characterized by time-changes in

*q*

_{m}(

*t*; Δ

*T*) and

*I*

_{m}(

*t*; Δ

*T*) at various quench depth Δ

*T*. When the quantities q

_{m}(

*t;*Δ

*T*) and

*t*are scaled, respectively, with the temperature-dependent characteristic parameters of the system

*q*

_{m}(0; Δ

*T*) and

*t*

_{c}(Δ

*T*), the scaled quantities

*Q*

_{m}(

*τ*) and \(\tilde I_{\rm{m}} (\tau )\), defined by

*T*, if Δ

*T*changes only the spatial scale via

*q*

_{m}(0;ΔT) and temporal scale via

*t*

_{c}(Δ

*T*) but not the coarsening mechanism of the systems (

*scaling hypothesis*).

Under the shear flow Δ*T* depends on \(\dot \gamma\) so that Δ*T* should be replaced by \({\rm \Delta} T(\dot \gamma ).\) The quantities *q*
_{m} and *I*
_{m} in the quiescent state should correspond to those *q*
_{mz
} and *I*
_{mz
} along the neutral axis under shear flow. Then the scaled quantities \(\hat Q_{\rm{m}} (\hat \tau )\) and \({\hat {\hskip -2pt I}}_{\rm{m}} (\hat \tau )\) under shear flow, which respectively correspond to *Q*
_{m}(τ) and \(\tilde {\hskip-2pt I}_{\rm{m}} (\tau )\) in the quiescent state, have been shown to give a universal behavior independent of \(\dot \gamma\) where \(\hat \tau\) is a reduced time under shear flow defined by \(\hat \tau \equiv t/t_{\rm{c}} ({\rm \Delta} T(\dot \gamma ))\) [44b]. Thus, *shear rate also changes spatio-temporal scale* of mixtures through the shear-rate-dependent Δ*T*.

### 5.5 Small Molecules Versus Polymers^{1}

The preceding section experimentally elucidated the shear-induced phase transition. The critical shear rate \(\dot \gamma _{\rm{c}}\) for the transition is assessed by observing the shear-rate dependence of the parameters that characterize the structure factor along the *q*
_{
z
} axis. These parameters are *I*(0)_{d} in the SQL function or integrated scattered intensity in the two-phase state and *I*(0)_{fl} and (*ξ*
_{⊥})_{fl} in the OZ function in the single-phase state.

*T*(defined by Δ

*T*10) in

*Figure*

*8-33*is brought into a single-phase system at \(\dot \gamma > \dot \gamma _{\rm{c}} ({\rm \Delta} T)\) as schematically shown in

*Figure*

*8-34a*. If one can measure \(\dot \gamma _{\rm{c}}\) as a function of

*T*below

*T*

_{c}(0), the critical temperature at \(\dot \gamma = 0,\) one obtains

*T*versus \(\dot \gamma _{\rm{c}} (T)\) as schematically shown in

*Figure*

*8-34b*.

*Figure*

*8-34b*also predicts a critical temperature drop from

*T*

_{c}(0) to

*T*by an amount of \({\rm \Delta} T_{\rm{c}} (\dot \gamma )\) ≡

*T*

_{c}(0) \(- T_{\rm{c}} (\dot \gamma )\) under a given shear rate of \(\dot \gamma = \dot \gamma _{\rm{c}}\) (see

*Figure*

*8-34b*or

*8-33*). The value \({\rm \Delta} T_{\rm{c}} (\dot \gamma )\) found experimentally was of the order of 1 mK for the small-molecule system [78] and 10 K for the polymer systems over the shear rate range experimentally covered [40, 61]. A law on the shear-rate dependence of the critical temperature drops \({\rm \Delta} T_{\rm{c}} (\dot \gamma )\) was first found to be given by (40) below for a small molecule system by Beysens et al. [78] and to be given by (41) for the symmetric polymer systems by Hashimoto and coworkers. [40, 61]

*Section*

*5.4*.

These laws are quite similar except for a very large difference in the prefactor of the shear-rate dependent term \(\dot \gamma ^{\rm{n}}\). The prefactor is the characteristic time *τ*
_{s} to the power 0.53 for the small-molecule system and the characteristic time \(\tau _{\rm{p}}\) to the power 0.5 for the polymer system. Since *τ*
_{p} ≫ τ_{s} [11, 21], it is quite reasonable, though striking, that the prefactor for the polymer system is very much larger than that of the small-molecule system *by a factor of about 10*
^{
4
}. *Polymer systems are much more sensitive to shear flow than small molecule systems*. This is quite natural from the following view points: (1) *Γ*
_{conc} for polymers \((\tau _{\rm{p}} ^{ - 1} )\) is much smaller than *Γ*
_{conc} for small molecules \((\tau _{\rm{s}} ^{ - 1} )\); (2) shear flow is effective when \(\dot \gamma > \Gamma _{{\rm{conc}}}\) as discussed earlier in *Section*
*2.1*
.

Based on the general principle introduced in *Section*
*2.1*
, the law for \({\rm \Delta} T_{\rm{c}} (\dot \gamma )\) can be qualitatively explained as follows. In the case when \(\dot \gamma < \Gamma _{{\rm{conc}}}\), phase separation proceeds and domains grow over the time period of order of \(\dot \gamma ^{ - 1}\). In the time scale longer than this, the shear deforms and destroys overgrown domains into small domains but destroyed small domains tend to grow again, driven by the thermodynamic driving force for the growth, giving rise to shear rate-dependent steady-state domain structures.

*Γ*

_{conc}. This concept together with the relation given by the physics of critical phenomena, \(\Gamma _{{\rm{conc}}} \sim {\rm{{\rm \Delta} }}T^n ,\) predicts that

The (42*a*
) predicts the critical shear rate as a function of quench depth \({\rm{{\rm \Delta} }}T = T_{\rm{c}} - T\), while (42*b*
) predicts the critical temperature drop \({\rm{{\rm \Delta} }}T_{\rm{c}} (\dot \gamma )\) as a function of shear rate. If the systems under consideration belong to 3D Ising universality class, the value *n* is equal to 1.93. If the systems belong to the mean-field universality class, the value of *n* is equal to 2 [30, 79]. These values of *n* together with (42) predict (40) and (41), respectively.

Han and his coworkers reported the relationship between the critical temperature drop and shear rate for deuterated PS and PB mixtures by means of dynamic light scattering and SANS with and without shear flow for the single-phase mixture [63, 65]. They found that their result is consistent with a renormalization group theory proposed by Onuki–Kawasaki [80–82].

Hashimoto and coworkers [83–85] reported that the spinodally decomposed symmetric polymer mixture of PB and SBR having \(M_{\rm{n}} \cong 10^5\) is brought into a single phase state near the critical point under a repeated uniaxial compression at 25°C which is designated by *Baker’s Transformation*. The mixture was estimated to have *T*
_{c} (0) ≥ 170°C in the quiescent state. The Baker’s Transformation effectively induces a shear flow at a small shear rate such that \(\dot \gamma < \Gamma _{\rm{d}}\), i.e., polymer chains under shear flow are in relaxed state in terms of orientation. Even at the small shear rate it satisfies the condition of \(\dot \gamma > \Gamma _{{\rm{conc}}}\) so that the mixture can be bought into shear-induced single-phase state. Since *Γ*
_{conc} is so small and hence \(\tau _{\rm{p}} ^{1/2}\) (or prefactor in (41)) is so large that even the small \(\dot \gamma\) is sufficient to drop the critical temperature by \({\rm{{\rm \Delta} }}T(\dot \gamma ) \ge 145^\circ {\rm{C}}\).

### 5.6 Tracing Back the Growth History of Phase-Separated Structures

*spinodal ring*which has an intensity maximum at

*q*=

*q*

_{m}, independent of the azimuthal angle (circular spinodal ring) [77].

*Figure*

*8-35*shows the time evolution of the characteristic length \(\Lambda _{\rm{m}} (t) \equiv 2{\rm{\pi }}/q_{\rm{m}} (t)\) for PS/PB (50:50)/DOP 3.0 wt% at Δ

*T*(0) = 6 K. In this figure the values of Λ

_{m}(

*t*) are given on the ordinate scale in the right-hand side of the figure, while those of

*t*can be found on the abscissa on the top. The

*t*-scale was deliberately chosen such that the value increases from right to left.

The log Λ_{m}(*t*) versus log *t* plot shows the coarsening of the domain structure as revealed by increase of Λ_{m}(*t*) with *t* such that the slope of the curve in the plot approximately changes 1/3 → 1/2 → 1/4 with time [44b]. The earlier part of the curve showing the change in the slope from 1/3 to 1/2 was elucidated to correspond to a coarsening process of bicontinuous phase-separated domains rich in the PS solution and those rich in the PB solution, while the later part of the curve showing the change in the slope from 1/2 to 1/4 was elucidated to correspond to the percolation-to-cluster transition (PCT) [21, 95] and a corresponding slowing down of the coarsening process after the PCT. After a completion of the PCT, it is expected that clusters of droplets grow with *t*
^{1/3}, though this time domain was not yet attained by this experiment.

It is important to note an intriguing possibility that increasing shear rate enables us to trace back the history of phase separation process of dynamically symmetric systems that have been already phase-separated in a quiescent state. This is because the shear rate determines the time available for the phase-separated domains to grow. Consideration should be given to the domain size along the neutral axis (the *oz*-axis or along the *q*
_{
z
} axis), because the deformation effect on the domain size is less significant along the neutral axis than along the flow axes (the *ox*-axis or the *q*
_{
x
} axis). Thus it may be possible to compare Λ_{m}(*t*) versus *t* obtained for the quiescent solution with (*ξ*
_{⊥})_{d} versus \(\dot \gamma\) obtained for the nearly same solution under steady-state shear by taking into account the fact that \(\dot \gamma\) should correspond to *t*
^{−1}.

*Figure*
*8-35*
demonstrates validity of the above conjuncture where log (*ξ*
_{⊥})_{d} versus log \(\dot \gamma\) also is plotted together with log Λ_{m}(*t*) versus log *t* on the same double logarithmic scale. The scales in the left ordinate and bottom abscissa should be referred to log (*ξ*
_{⊥})_{d} and log \(\dot \gamma\), respectively. It should be noted that the top abscissa corresponds to log *t* = (constant) – log \(\dot \gamma\) where the constant was chosen such that the two curves are overlapped. The fact that the concentration and Δ*T* employed for the two experiments are slightly different should not alter the self-assembling process itself but slightly alter the time and spatial scales of the two experiments. Thus the two curves are shifted both vertically and horizontally in the double logarithmic scale. Upon shifting, the two results are demonstrated to well-superpose each other, revealing the *equivalence principle* of \(\xi _{ \bot d} (\dot \gamma )\) and Λ_{m}(*t*) under a condition of \(\dot \gamma \sim t^{ - 1}\). Thus in principle one can trace back of a growth history of the phase-separated domain structures at a given stage of observation upon increasing shear rate for the particular cases as specified earlier!

### 5.7 Further Remarks

The author would like to point out a series of interesting works done by Mewis and coworkers on binary systems at low shear rates (corresponding to Regime III in *Figure*
*8-27*
). They elucidated that SALS is capable of discriminating between the different modes of relaxation for droplets: retraction, end-pinching, and breakup by Rayleigh instabilities [56]. The pattern evolution was modeled by means of a diffraction approach [57]. The proposed model together with the Tomotika theory are demonstrated to provide an elegant method of estimating the interfacial tension as low as 10^{−6} N/m for immiscible blends. They demonstrated that SALS can be used to unambiguously determine the breakup time of droplets during shear flow [96]. Systematic measurements of this breakup time led to a nice scaling law [97].

## 6 Shear-Induced Demixing (Phase Separation)

As discussed earlier, the stress–diffusion coupling or the viscoelastic effects in dynamically asymmetric systems suppresses the relaxation rate [18] or growth rate [17] of concentration fluctuations. This kind of suppression is anticipated to cause the asymmetric systems to become increasingly susceptible to a weak shear with increasing asymmetry parameter *α*
_{a} (6).

### 6.1 Observation of Shear-Induced Dissipative Structures

When a semidilute solution in a single-phase state is subjected to shear flow with a shear rate greater than a critical value, the solution was found to change from a transparent solution to a turbid solution [98]. It was also reported that this shear-induced turbidity brings a related change in the rheological behavior to the solution [86, 99, 100]. This intriguing phenomenon, as elucidated from transmitted light intensity and rhological properties as a function of shear rate \(\dot \gamma\), suggests that the shear flow induces formation of a dissipative structure in the solution, reflecting either shear-enhanced concentration fluctuations or shear-induced phase separation. It is important to note the following point. The shear-induced phase separation at a given temperature implies that the shear flow elevates the critical temperature. The increased critical temperature brings about an initially single-phase solution into a phase-separating solution.

*Sections*

*2*and

*4*. Shear small-angle light scattering (shear-SALS) [1, 2, 101–105] and shear small-angle neutron scattering [106, 107] elucidated the scattering structure factor for systems under shear flow, while shear optical microscopy (shear-OM) elucidated real-space structures [31, 32, 108–110].

*Figure*

*8-36*represents typical experimental results on shear-SALS and shear-OM images in the

*x-z*plane for PS548/DOP 6.0 wt%, a typical system having the dynamical asymmetry. The system has the cloud point at 13.8°C [32, 108, 109]. The experiments were done in the single-phase region at 27°C, 13.3°C above the cloud point.

The solution at rest (at \(\dot \gamma = 0\;{\rm s}^{ - 1}\)) is in a single-phase state so that it does not exhibit appreciable SALS at the small angles as shown in part (a) and no features in the OM image as shown in part (e). The halo around the direct beam stop (appearing as a dark circle) in part (a) is still a tail of incident beam; the true scattering is very weak so that it is buried in the dark background in this pattern. Upon imposing shear flow of \(\dot \gamma = 0.23\;{\rm s}^{ - 1}\), strong scattering appears along the flow direction but scattering along the neutral direction (horizontal direction in the figure) remains almost unchanged from that for the quiescent solution so that the scattering exhibits so called *butterfly pattern* with the butterfly wings spreading along the flow-direction and with the dark string appearing along the neutral axis, as shown in part (b).

The OM images shown in part (f) to (h) exhibit some anisotropic contrast variations with a characteristic length along the flow direction of the order of 10 μm which may reflect dissipative structures developed by shear-induced phase separation or concentration fluctuations. The fact that the structures observed under the OM images reflects true structures developed in the sheared solution was confirmed by comparing the first Fourier transform (FFT) pattern of the image with the real-scattering patterns. For example the FFT patterns of the images in parts (e) and (f) are respectively shown in parts (c) and (d). One can judge that the pattern (d) exhibits characteristics of the butterfly pattern shown in the pattern (b) and that the patterns (a) and (c) consistently represent a homogeneous solution.

The butterfly patterns were observed at various temperatures and concentrations [111] and for various solvents [112] such as cyclohexane and diethyl malonate which are *θ* solvent for PS at 35°C as well as dibutyl phthalate and tricresyl phosphate which are good solvents for PS. They were also observed for semidilute solutions of polyethylene in paraffin as a solvent (athermal solution) [110, 113, 114] and for sheared colloidal suspensions [115]. They are thus quite general for sheared dynamically asymmetric systems.

It should be noted that these shear-induced concentration fluctuations and/or phase-separation (demixing) were also found for bulk polymer mixtures. The shear-induced demixing was reported for PS/PVME mixtures by Winter and coworkers [71] and Higgins and coworkers [116]. The shear-induced demixing was found also for other polymer mixtures [73–76].

### 6.2 Origin of Shear-Induced Formation of Dissipative Structures

*Figure*

*8-37*represents schematically the origin of the butterfly-type scattering [11, 103]. The concentration fluctuations of the single-phase system in quiescent state are small in amplitude and isotropic as schematically shown in part (a). Consequently the corresponding OM image is featureless, and the scattering is weak and circularly symmetric around the incident beam axis (

*y*-axis set normal to the plane of the paper) as shown in part (b). The shear flow at shear rates larger than the critical shear rate \(\dot \gamma _{{\rm{c,x}}}\) enhances plane-wave type concentration fluctuations along the flow direction (

*x*-axis) but not much along the neutral direction (

*z*-axis), as shown in part (c).

These shear-enhanced concentration fluctuations give rise to strong scattered intensity along the *x*-axis but not much along the *z*-axis, giving rise to the butterfly-type anisotropic scattering pattern, as shown in part (d), and the contrast variation of the transmission OM image, as shown in part (c). Upon increasing \(\dot \gamma\), the plane wave fluctuations grow in the directions away from the *x*-axis also so that the butterfly wings are spread over a wider azimuthal angle *μ* and the OM image undergoes corresponding changes [108]. However, the intensity along the *z*-axis is kept unchanged from that for the quiescent solution up to a certain critical shear rate \(\dot \gamma _{{\rm{c,}}z}\), resulting in the dark streak or sector along the *z*-axis in the butterfly pattern. The shear rate \(\dot \gamma _{{\rm{c,}}z}\) is the critical shear rate above which the scattering intensity increases even along the *z*-axis [111], as will be detailed later.

*Figure*

*8-38*schematically shows a “solvent-squeeze model” for the shear-enhanced concentration fluctuations [117]. The single-phase solution in the quiescent state is subjected to thermal concentration fluctuations and thus has regions rich in polymers having more entanglements as shown in the gray regions of part (a) and regions poor in polymers having fewer entanglements (bright matrix). Under imposed shear flow, the regions rich in polymers are accompanied by larger stress than the other region through concentration dependence of viscosity

*η*and the coefficeient of first normal stress difference

*ψ*

_{1}, giving rise to local stress variations. The stress developed under shear flow can be relaxed via disentanglements, in the case when \(\dot \gamma\) is smaller than the disentanglement rate

*Γ*

_{dis}. In this case, the concentration fluctuations can be also relaxed, and the solution remains homogeneous even under shear flow.

However in the case when \(\dot \gamma > \Gamma _{{\rm{dis}}}\), the stress developed in deformed swollen entangled networks can not have sufficient time to be relaxed by disentanglements. As a consequence, the stress or deformation and orientation of the entangled polymers can be relaxed only by squeezing solvents from the regions rich in polymers, having higher elastic energy, into those poor in polymers against osmotic pressure as shown in part (b), resulting in regions rich in polymer becoming even richer and those poor in polymer becoming even poorer. Upon squeezing, swollen deformed chains can have relaxed conformations, hence resulting in free-energy release and stress relaxation. Since deformations of swollen entangled chains are anisotropic, the solvent-squeeze is also anisotropic, and hence the shear-enhanced concentration fluctuations become also anisotropic. The Fourier modes of the shear-enhanced fluctuations with wave vector **q** spread over a limited range in azimuthal angle *μ* in the *x-z* plane, as shown in *Figures*
*8-37c*
and *d*.

### 6.3 Shear-Rate Dependence

*Figure*

*8-39*represents typical shear-rate dependence of steady-state structures as observed by SALS and rheological properties for PS548/DOP 6.0 wt% at 27°C [33]. The figure shows coefficient of the first normal stress difference

*ψ*

_{1}, shear viscosity

*η*, normalized integrated scattered intensity parallel and perpendicular to flow \({\mathcal {i}}_\parallel (\dot \gamma )/{\mathcal {i}}_\parallel (\dot \gamma = 0)\) and \({\mathcal {i}}_ \bot (\dot \gamma )/{\mathcal {i}}_ \bot (\dot \gamma = 0)\), respectively, as a function of \(\dot \gamma\) in a double logarithmic scale. Here the integrated scattered intensity \({\mathcal {i}}\!_j (\dot\gamma) (\,j=\, \parallel {\rm or} \ \bot )\) is defined as

*I*(

*q*

_{ x },

*q*

_{ z }) is the scattered intensity distribution in the (

*q*

_{ x },

*q*

_{ z }) plane. In the case of \({\mathcal {i}}_\parallel (\dot \gamma )\), quantity

*b*(34 × 10

^{−5}nm

^{−1}) and –

*b*are the upper and lower limits of the integration along

*q*

_{ z }, respectively, and

*a*

_{2}and

*a*

_{1}are those along

*q*

_{ x }(

*a*

_{2}= 2.7 × 10

^{−3}nm

^{−1},

*a*

_{1}= 6.2 × 10

^{−4}nm

^{−1}). \({\mathcal {i}}_j (0)\) is \({\mathcal {i}}_j (\dot \gamma )\) at \(\dot \gamma = 0\). The quantities

*a*

_{1},

*a*

_{2}, and

*b*for the case of \({\mathcal {i}}_ \bot (\dot \gamma )\) are similarly defined.

*Figure*
*8-40*
shows the steady-state scattering patterns taken at the shear rates indicated by the annotations *a* to *d* in *Figure*
*8-39*
. The results displayed in *Figure*
*8-39*
might give us such an impression that the power law behaviors of rheological properties *η* and *ψ*
_{I}, representing the shear thinning, arise from the shear-enhanced concentration fluctuations as revealed by the shear-enhanced scattering. However, this is not always the case, as is obvious in polymer melts which exhibit the similar shear thinning but no shear-enhanced scattering. Here we intend to compare the rheological and scattering behaviors obtained for the same sample specimens on the same graph. In conjunction with this figure, we would like to raise an important question on whether or not or how the concentration fluctuations affect the rheological behaviors [118].

The scattering and rheological behavior as a function of shear rate can be classified into four regimes as shown in *Figure*
*8-39*
. In Regime I the concentration fluctuations or the scattering intensity are essentially identical to those in the quiescent solution as shown by the integrated scattered intensity and the SALS pattern in *Figures*
*8-39*
and
*8-40a*
, respectively. Here *η* is nearly independent of \(\dot \gamma\). In this regime, \(\dot \gamma\) is smaller than the relaxation rate of the concentration fluctuations *Γ*
_{conc} or disentanglement rate so that shear flow would not essentially affect the fluctuations, as discussed earlier in *Section*
*2.1*
or *Section*
*6.2*
.

Regimes II–IV exist at \(\dot \gamma > \dot \gamma _{c,x}\), a critical shear rate above which shear-enhanced scattering is observed along the *x-*axis together with the shear thinning as evidenced by the decrease of *η* and *ψ*
_{1} with \(\dot \gamma\). In this regimes \(\dot \gamma > \Gamma _{\rm conc}\) so that shear flow affects the concentration fluctuations and hence the thermodynamic state of the solution. As discussed in *Section*
*3.2*
the system under considerations is extremely dynamically asymmetric so that the stress–diffusion coupling would strongly suppress *Γ*
_{conc} as discussed in *Section*
*3.3*
. As a consequence, the shear flow with even small shear rates can enhance the concentration fluctuations via the solvent-squeeze mechanism encountered in the stress relaxation process of deformed entangled polymer coils, as elaborated in *Section*
*6.1*
.

In Regime II, the scattering intensity dramatically increases along the flow direction such that \({\mathcal {i}}_\parallel(\dot \gamma ) / {\mathcal {i}}_\parallel (0)\) increases by approximately 30 fold as shown in *Figure*
*8-39*
. However the intensity normal to flow \({\mathcal {i}}_ \bot\) approximately remains that in the quiescent solution, giving rise to the butterfly pattern as shown in *Figure*
*8-40b*
. Therefore the shear-enhanced concentration fluctuations may be approximately given by plane-wave-type concentration fluctuations with their wave vectors preferentially oriented paralled to the flow direction. *ψ*
_{1} and *η* starts to decease with increasing \(\dot \gamma\). In Regime III, \({\mathcal {i}}_\parallel(\dot \gamma ) / {\mathcal {i}}_\parallel (0)\) increases upto ∼100 fold and keeps the high intensity level with \(\dot \gamma\), while \({\mathcal {i}}_ \bot\) starts to increase and reaches a constant value with increasing \(\dot \gamma\) above the critical shear rate of \(\dot \gamma _{{\rm{c}},z}\). This increase of the scattering intensity and hence of the concentration fluctuations along the neutral or vorticity direction is a nonlinear effect brought about by the further enhancement of concentration fluctuations along the flow direction. It may be reasonable to attribute this nonlinear effect to onset of the shear-induced demixing (phase separation) [107, 111, 119]. In this regime *ψ*
_{1} and *η* decrease with \(\dot \gamma\) according to power laws. Consequently dissipative structures developed under flow tend to minimize mechanical energy dissipation in this nonequilibrium process. The scattering pattern in *Figure*
*8-40c*
and the OM images in *Figure*
*8-36f*
*–*
*h*
belong to this regime.

In Regime IV at shear rates larger than the critical shear rate of \(\dot \gamma _{\rm{a}} ,\) anomalies are observed in both the scattering and the rheological properties. The scattering anomaly is seen in a further increase of both \({\mathcal {i}}_\parallel\) and \({\mathcal {i}}_ \bot\) by a factor of ∼5 and ∼10, respectively, as shown in *Figure*
*8-39*
as well as in the newly appeared strong streak-like scattering patterns along *q*
_{
z
} axis as shown in *Figure*
*8-40d*
. This anomaly is accompanied also by rheological anomalies. Upon increasing \(\dot \gamma\) from a value in Regime III to that in Regime IV, *the shear stress and normal stress difference momentarily increase dramatically*. This process must give rise to a high degree of polymer chain orientation as manifested by the appearance of large negative birefringence as will be elaborated later in conjunction with *Figure*
*8-42*
. However if the solution is kept sheared, they eventually decay with time to reach steady values, though small variations still exist around the steady values. The steady values of *η* and *ψ*
_{1} in Regime IV are larger than those in Regime III, and they tend to increases with \(\dot \gamma\).

*Figure*

*8-41*represents typical steady-state SALS patterns for the two different systems in Regime IV (part

*a*and

*b*) as well as an OM image (part

*c*) obtained for the solution showing the SALS pattern in (part

*b*) [51]. The SALS patterns (a) and (b) in this regime show a strong streak along the neutral direction (the

*q*

_{ z }axis) superposed on the butterfly pattern. The OM image in (c) shows an image of string-like structures along the flow direction (the

*x*-axis) which is expected to correspond to the strong streak-like scattering pattern along the neutral direction in (a) and (b). These string-like structures have a large optical anisotropy as evidenced by the large negative birefringence as described below [120].

*Figure*
*8-42*
summarizes a result obtained by the rheo-optical studies, concerning SALS, OM, form optical dichroism, and birefringence, and dissipative structures developed under steady shear flow as a function of \(\dot \gamma\)[120]. At \(\dot \gamma < \dot \gamma _{c,x}\) (in Regime I) the sheared solution is essentially identical to homogeneous quiescent solutions, as evidenced by no-SALS pattern and no characteristic OM image as shown in part (a), as well as no form optical dichroism and no birefringence. At \(\dot \gamma > \dot \gamma _{{\rm{c}},x}\) shear-induced dissipative structures are formed due to the elastic effects mediated by the dynamical asymmetry effects and consequent stress–diffusion coupling effects in sheared solutions where entangled polymer coils are deformed as a consequence of essentially no time available for disentanglements of polymer chains under shear flow.

In Regime II at \({\dot \gamma}_{{\rm c},x} < \dot \gamma < \dot \gamma_{{\rm c},z}\), the butterfly scattering appears but the scattering intensity along the neutral axis (the *z*-axis) remains as weak as that from the quiescent solution, and OM images show dark and bright contrast variations along flow direction as shown in part (b). Thus it is expected that shear-enhanced concentration fluctuations along the flow direction are built-up but the fluctuations along the neutral axis stay at the same level as those in the quiescent solution. In Regime III the butterfly pattern is further developed in such a way that the butterfly wings spread over a wider range of azimuthal angle (*μ*), and that the scattering intensity along the neutral direction increases also. The OM images show more clearer contrast variations (see part c) than those in Regime II, indicating evolution of domain-like structures extended along the neutral axis as schematically shown in part (d). These facts may imply that the phase separation occurs in Regime III, giving rise to phase-separated domains with the average spacing along *q*
_{
x
} of the order of \({\rm{2\pi }}/q_{{\rm{mx}}} = 10\;{\rm \mu} {\rm m}\), but centers of mass of individual domains being randomly placed along the neutral axis.

The domains tend to grow due to thermodynamic instability of solution driven by the elastic effects. However, the overgrown domains tend to be burst due to the elastic force and viscous drag, resulting in formation of shear-rate dependent steady-state dissipative structures. In Regime II and III, the dichroism is negative and still has small absolute values, consistent with the domains extended and oriented along the neutral axis. Birefringence has also small negative values, revealing that PS chains are only weakly oriented along the flow direction.

In Regime IV, the (phase-separated) domains developed in Regime III tend to align, driven by hydrodynamic interactions, with their centers of mass along the flow direction, resulting in the string-like assembly of the domain structures. As a consequence, SALS shows a strong streak along the neutral axis. The butterfly pattern appearing in superposition of the streak pattern may reflect the scattering from the domains comprising the string with an average interdomain distance of order of 10 μm. The domains themselves cannot be clearly resolved in *Figures*
*8-41c*
and
*8-42e*
and is inferred from the butterfly scattering patterns observed in *Figure*
*8-41a*
and *b*. This dissipative structure and the SALS pattern developed are quite *reminiscent of the pearl-necklace structure* and its corresponding SALS pattern shown in *Figure*
*8-30*
. The dissipative structure, having the string-like alignment of the phase-separated domains parallel to the flow axis, is anticipated to be a rational optimum structure which may minimize mechanical energy dissipation under such a strong shear flow.

As a consequence of the string-like structure, the form dichroism shows a large positive value. The birefringence shows a large negative values, indicating that end-to-end vectors of PS chains in the strings are oriented parallel to the string axis, i.e., along the flow direction. *Thus the strings have a large optical anisotropy*. More precisely, the strings developed in Regime IV exhibit a sharp transition from a weak optical anisotropy to a strong optical anisotropy with increasing \(\dot {\gamma}\). Moreover, this optical transition occurs in parallel to the rheological transition from the shear thinning behavior to the shear thickening behavior as shown in *Figure*
*8-39*
. These transitions may indicate formation of bundles of oriented chains interconnecting the demixed domains in the strings in a narrow shear rate range.

The optically anisotropic strings having the bundles of oriented chains developed in the PS solutions are strikingly similar to those developed in the sheared semidilute solutions of polyethylene, as will be discussed later in section 6.6. However the PS used is not a crystallizable polymer so that crystallization would not occur in Regime IV. However, if the polymers used were crystallizable, they would be crystallized into fibrillar crystalline superstructures of the so-called “shish-kebab” structures. This is well-known as shear-induced crystallization of polymer solutions. Therefore, it would be extremely important to recognize the possibility that this shear-induced crystallization may be triggered by the shear-enhanced concentration fluctuations and/or phase separation, at least in the semidilute solutions of interest here, simply because crystallization is anticipated to occur in the regions rich in polymer concentrations. This point will be briefly discussed further in *Section*
*6.6*
later.

### 6.4 Time-Evolution of Transient Dissipative Structures

The preceding section discussed various dissipative structures developed at steady-state shear flow for a range of shear rates. The discussion now focuses on how transient structures are developed and grown into the steady-state structures after onset of a step-up shear flow from \(\dot \gamma = 0\) to a shear rate larger than the critical shear rate, i.e., \(\dot \gamma > \dot \gamma _{{\rm c,z}}\) or \(\dot \gamma > \dot \gamma _{\rm a}\).

*Figure*

*8-43*shows the transient rheological properties, the first normal stress difference

*N*

_{1}and shear stress

*σ*, for the PS548/DOP 6.0 wt% solution at 27.5°C after the onset of the step up shear flow of a constant shear rate \(\dot \gamma = 14.3\;{\rm s}^{ - 1}\) in Regime III defined in

*Section*

*6.3*(see

*Figures*

*8-39*and

*8-42*) [33]. Both

*N*

_{1}and

*σ*clearly showed two stress-overshoots at least, and even the third overshoot may be discerned at around 70 s on its approach toward steady-state. Each overshoot in

*σ*appears to occur prior to the corresponding one in

*N*

_{1}. The structural origin of this intriguing rheological behavior was simultaneously followed up by the shear-SALS method. The results are shown in

*Figure*

*8-44*where the patterns (a) and (b) (both taken at

*t*= 0 s) as well as the patterns (m) and (n) (both taken at

*t*= 150 s) represent the original pattern and the contour pattern, respectively, for comparison.

The scattering intensity along the flow direction started to remarkably increase with time at about 15 s (pattern e), and then after that time the butterfly pattern was clearly developed. Its intensity parallel to the flow direction increases with time, while the intensity normal to the flow direction stays weak up to about 30 s, almost at the same intensity level as that for the quiescent solution, giving rise to the dark streak as discussed in the preceding section. The width of this dark streak around the neutral axis narrowed with time. The results indicate that the amplitude of the concentration fluctuations increases with time and the Fourier modes of the fluctuations spread over a wider azimuthal angle *μ* (see *Figure*
*8-37*
for the definition of *μ*) with respect to the flow direction. The intensity normal to the flow direction tends to increase with time beyond 30 s after the onset of the flow, implying the onset of the shear-induced phase separation. Thus transient changes in the scattering patterns with time seem to be quite similar to the change in the steady-state scattering patterns with shear rate.

The butterfly pattern had a scattering maximum along the flow direction at the scattering vector *q*
_{max}, its value slightly shifting with time (from ∼0.6 × 10^{−3} nm^{−1} to ∼0.35 × 10^{−3} nm^{−1}) over the *q*-range covered in this experiment. Thus the characteristic length \(2{\rm{\pi }}/q_{\max }\) of the shear-induced structure along the flow direction varies from ∼10 to 18 μm with time. From right after the onset of the flow (0 s) to 50 s, the scattered intensity increased by more than 1 order of magnitude [33]. The small-angle scattering patterns, especially the intensity parallel to the flow direction, showed no appreciable changes with time beyond 50 s, indicating the global dissipative structure reached a steady-state.

Comparisons of the two results shown in *Figures*
*8-43*
and
*8-44*
reveal the following pieces of evidence. The butterfly patterns are clearly developed and hence shear-induced structures (or concentration fluctuations) appear to be clearly formed after the first overshoot of *N*
_{1} and *σ* at around the time when the stress level is near the minimum in between the first and second overshoots. As time further elapses, both *N*
_{1} and *σ* increase toward the maximum values at the second overshoot; the butterfly pattern concurrently increases its scattered intensity and the butterfly wings spread over a wider azimuthal angle *μ*. Thus the rise of the stress level of *N*
_{1} and *σ* with time after the first stress overshoot is related to the growth of the characteristic structures discussed in the preceding paragraph. At around the time when *N*
_{1} and *σ* reach or slightly pass through the peak of the second overshoot (point (i) in *Figure*
*8-43*
), the demixing may start to occur, which may be related to the increase of the intensity along *q*
_{
z
}. The demixing appears to decrease the stress levels of *N*
_{1} and *σ* toward their steady values. As the stress levels approach the steady values, the global structures also appear to attain the steady structures at around point (k) in *Figure*
*8-43*
. The steady structures themselves were discussed in the preceding section (see *Figure*
*8-42*
). However it may be well-conceivable that the stress levels and the local SALS intensity *fluctuate* or even *oscillate* around their steady values, because demixed structures will grow under flow but over grown structures will be raptured by the flow, and a dynamical balance between the growth and the destruction will give the local *fluctuations* or *oscillations* though they tend to be averaged out over the large region of observation.

The stress increase in the first stress overshoot process is expected to arise from uniform deformation and the resulting orientation of entangled polymer chains in the solution [121], as the SALS intensity showed no remarkable increase in intensity. The stress relaxation after the first stress overshoot process is expected to be due to the molecular relaxation process of deformed polymer chains [121]. However the relaxation process in this case involves also the squeezing of solvents from the regions rich in polymer concentrations and hence builds up the concentration fluctuations against osmotic pressure. This process is responsible for developing the butterfly pattern. The relaxation time for the solvent squeezing process was clearly elucidated by large amplitude oscillatory shear flow (LAOS) experiments on the same semidilute solutions [117]. In this experiment, the butterfly pattern was observed at strain amplitudes *γ*
_{0} greater than a critical value *γ*
_{0,c}. At γ_{0} > γ_{0,c}, the butterfly pattern was observed at angular frequency *ω* satisfying \(\omega _{{\rm{c}},1} < \omega < \omega _{{\rm{c,u}}}\), where \(\gamma _0 \omega _{{\rm{c,l}}}\) corresponds to \(\dot \gamma _{c,x}\) (and hence to *Γ*
_{conc}) and \(\gamma _0 \omega _{{\rm{c,u}}}\) corresponds to the rate for the solvent squeezing [117]. At \(\omega \ge \omega _{c,u}\) the entangled polymer solutions are uniformly stretched and relaxed under the oscillatory shear flow, giving rise to no concentration heterogeneities, simply because there would be no time available for solvents to be squeezed out in order to relax the stress for the deformed swollen entangled networks. The growth of the shear-induced structure and deformation of the grown structure under the flow would again increase in the stress levels, giving rise to the second overshoot.

*Figure*

*8-45*schematically summarizes time evolutions of SALS, OM images, and

*N*

_{1}after a step-up increase of \(\dot \gamma\) from zero to a value larger than \(\dot \gamma _a\). A series of events induced by the step-up shear flow, leading to the steady state, is summarized as follows [33]:

- 1.
Incubation period

*t*<*t*_{c,x }, part (a), corresponding to the period of the first stress overshoots where entangled polymer chains in solutions are uniformly deformed and the increased stress starts being relaxed by the molecular relaxation process of deformed polymer chains accompanied by the solvent squeezing process; - 2.
An early stage of structure formation

*t*_{c,x }<*t*<*t*_{c,z }, part (b), corresponding to the period between the first and second stress overshoot where the shear-induced concentration fluctuations with a characteristic length scale of ∼10 μm are observed along*q*_{ x }in the*q*_{ z }-*q*_{ x }plane due to the solvent squeezing process; - 3.
A late stage of structure formation at

*t*>*t*_{c,z }, part (c), in the case of \(\dot \gamma < \dot \gamma _a\), or parts (c) and (d) in the case of \(\dot \gamma > \dot \gamma _a ,\) corresponding to the period of the second stress overshoot being relaxed toward an average steady-state stress level where phase separation starts to occur and grow, overgrown phase-separated structures rapture, and the steady-state dissipative structures (the optically anisotropic string-like structures in the case of \(\dot \gamma > \dot \gamma _a\)) are eventually formed.

### 6.5 Further Remarks

*Section*

*6*, brief remarks are added on the following important topics that have not been covered so far:

- 1.
Effects of concentrations and temperature of the semidilute solutions on critical shear rate \(\dot \gamma _{{\rm c},x}\) [111]

- 2.
Effect of solvent quality on critical shear rate \(\dot \gamma _{{\rm c},x}\) [112]

- 3.
A combined LS and SANS observations of the dissipative structures over a wide

*q*-range on the semidilute solutions subjected to continuous shear flow [119] - 4.
Effects of large amplitude oscillatory shear flow (LAOS) on dissipative structures, as observed by a combined SALS and SANS [117, 122]

- 5.
A possible difference in the dissipative structures formed under continuous shear flow and LAOS [123]

- 6.
Shear-induced phase separation in nonentangled, dynamically asymmetric systems [124]

- 7.
Shear-induced structures for phase-separating semidilute solutions [125]

*K*

_{os}is the osmotic modulus defined by,

*v*

_{0}is the volume of a monomeric unit,

*η*

_{0}is the zero-shear viscosity,

*N*

_{n}is the number average degree of polymerization,

*ϕ*

_{0}is the volume fraction of polymer, and

*χ*is the thermodynamic interaction parameter between polymer and solvent. Consequently the effects of concentration, temperature, and solvent quality on \(\dot {\gamma}_{ex}\) can be predicted through those effects on

*K*

_{os}/η

_{0}. The critical temperature increase with \(\dot \gamma\) is quantitatively predicted by (44) derived from the HFMO dynamical equation [3–5] described earlier in section 2.1.

*M*

_{w}= 2.0 × 10) solution in DOP.

*Figure*

*8-46*presents typical combined scattering profiles parallel to flow (part a) and perpendicular to flow (part b) obtained at various \(\dot \gamma '\)s at a given temperature

*T*= 22°C in a single phase state above the cloud point (lower than 12°C) [119].

*q*-region, (2) the scattering function approximated by a power law

*q*-region at

*q*<

*q*

_{ x }

_{c}, and (3) the OZ scattering function from thermal concentration fluctuations of entangled polymer solutions which appears in the SANS

*q*-region at

*q*>

*q*

_{ x }

_{c}. The OZ is independent of \(\dot \gamma\) and identical to the OZ in the quiescent solutions shown by the dotted line, implying that the imposed shear flow does not affect at all the thermal concentration fluctuations of the system. Therefore, it is expected that the relaxation rates of the Fourier modes of thermal concentration fluctuations

*Γ*

_{conc}at

*q*

_{ x }>

*q*

_{ x }

_{c}and at

*q*

_{ z }>

*q*

_{ z }

_{c}are higher than \(\dot \gamma\) covered in this experiment, so that

*I*(

*q*

_{ x }) =

*I*(

*q*

_{ z }), both being independent of \(\dot \gamma\). The exponents

*α*in the power law region tend to increase with \(\dot \gamma\) from ∼1.6 to ∼3.

The scattering excess to OZ scattering is observed at \(\dot \gamma \ge 0.4\;{\rm s}^{ - 1}\) for SALS and \(\dot \gamma \ge 0.2\;{\rm{s}}^{ - 1}\) for SANS along *q*
_{
x
}. This means that the butterfly pattern is observed only by SANS at \(0.2\;{\rm s}^{ - 1} < \dot \gamma \; < 0.4\;{\rm s}^{ - 1}\), SALS still tends to look at the system as a homogeneous solution in this shear rate range. Thus one must note that the assessment of \(\dot \gamma _{{\rm{c,x}}}\) may depend on the *q*-range experimentally covered. However, at \(\dot \gamma \ge 0.4\;{\rm s}^{ - 1}\) the excess scattering is discerned over the SALS and SANS *q*-ranges, which reflects the shear-induced structure extending over a wide length scale. These are more clearly demonstrated in *Figure*
*4*
of [119] which directly compared the SALS and SANS patterns over a range of shear rates. The OZ scattering shows the power law behavior of (
*46*
) with exponent *α* equal to 2 at the high q-limits of *q*
_{
x
} and *q*
_{
z
} larger than 0.3 nm^{−1} as obviously seen in *Figure*
*8-46*
[119].

*γ*

_{0}, angular frequency

*ω*, strain phase

*ϕ*=

*ωt*, and azimuthal angle

*μ*. A kind of phase diagram for shear-induced concentration fluctuations or phase-separation was constructed in the parameter space of

*γ*

_{0}and

*ω*at a given temperature for the single-phase solution in the quiescent solution. At a given

*γ*

_{0}greater than critical value \(\gamma _{0,{\rm{c}}} \cong 2\), the shear–induced butterfly scattering pattern was observed in the frequency range given by

*Γ*

_{squeeze}is the rate of solvent squeeze from the deformed entangled polymer chains mediated by the elastic effects [122]. The range \({\rm \Delta} \omega = \omega _{\rm{u}} - \omega _{{\rm{ l}}}\) where shear-induced butterfly scattering observed tends to increase with

*γ*

_{0}. At

*ω*>

*ω*

_{u}, entangled polymer chains will be deformed and relaxed uniformly without the solvent squeezing process so that the solution is kept homogeneous. At

*ω*<

*ω*

_{1}, deformed entangled polymer chains can be relaxed via disentanglements so that the solution is also kept homogeneous.

The dissipative structures as observed by SALS along *q*
_{
x
} under the LAOS have two components; a stationary component independent of the strain phase and an oscillatory (or dynamic) component which is developed and relaxed with the strain phase. The time-evolution of the former part is due to purely nonlinear effects on the structure formation induced by LAOS. The characteristic length of the latter part of the shear-induced structures was elucidated to decrease with increasing *ω* for a given *γ*
_{0} (see *Figure*
*11*
of [117]).

*q*

_{ x }axis under LAOS also show a crossover from the power law behavior, as given by (

*46*), to the OZ scattering from the concentration fluctuations of the entangled polymer chains in the domains and matrix, as shown in

*Figure*

*8-47a*[122, 123]. The power law exponent

*α*increases with \(\dot \gamma\) from 2.5 and 3.4, the values of which are larger than those for the case of the continuous shear flow (

*Figure*

*8-46a*). After the subtraction for the OZ contribution, the power law scattering

*I*

_{d}(

*q*

_{ x }) tends to exhibit the exponent of 4 at all \(\dot \gamma\) ’s, as shown in

*Figure*

*8-47b*, indicative of domain structures with a sharp interface boundary. On the other hand in the case of the continuous shear flow, the power law after the correction for the OZ contribution exhibits the exponent from 2.4 to 3 with \(\dot \gamma\), never reaching 4. It is worth noting that the crossover

*q*value,

*q*

_{ x }

_{c}(∼0.1 nm

^{−1}) in

*I*(

*q*

_{ x }) in LAOS is higher than that in continuous shear flow (

*q*

_{ x }

_{c}= 0.05 nm

^{−1}).

A comparison of the two pieces of evidence described earlier, regarding the shear-induced SANS observed under the continuous shear flow and LAOS, addresses a fundamental question raised by item (5) in the beginning of this section: LAOS can drive the semidilute solutions into phase-separated state with microdomains having sharp interfacial boundaries, while the continuous shear flow does not. For the continuous shear flow, interfacial boundaries remain less well-defined for all the shear rates covered in this experiment. This important experimental discovery was qualitatively interpreted [123] in the context of the HFMO theory [3–5].

Below, a very brief explanation is provided about the important difference described earlier. Further details were given elsewhere [123]. Under continuous shear-flow, *q*-Fourier modes of the concentration fluctuations are convected (i.e., rotated and stretched or compressed depending on their orientation). They are enhanced for the growth in the first and third quadrants and suppressed in the second and fourth quadrants in the *q*
_{
x
}-*q*
_{
y
} plane; this acts as a stabilizing mechanism of the solution against indefinite growth of the fluctuations. However, under oscillatory shear flow all Fourier modes of the fluctuations oscillate. A Fourier mode of the fluctuations which is initially oriented near the *q*
_{
x
} axis will oscillate about *q*
_{
x
} and hence will repeatedly pass near the *q*
_{
x
} axis where the enhancement of fluctuations by normal stress is strongest. Thus the convective stabilizing mechanism is absent, and hence LAOS can drive the system all the way to form phase-separated domains with sharp interfacial boundaries, consistent with experimental observation. All other Fourier modes, which will not pass the *q*
_{
x
} axis in the oscillating shear flow, are subjected to shear forces which alternately enhance and suppress the growth of the fluctuations. Thus fluctuations should not be enhanced as much by shear stress under oscillatory flow as under continuous flow. This is why the butterfly patterns in the *q*
_{
x
}-*q*
_{
z
} plane are most intense along the *q*
_{
x
} direction.

So far, shear-induced structures for dynamically asymmetric systems with entanglements have been discussed. The item (6) concerns shear-induced structures for a nonentangled system having a dynamical asymmetry due to a disparity of glass transition temperatures of two components in the blend. The detailed discussions on this item were described elsewhere [124].

To this point, shear-induced dissipative structures for a single-phase solution in quiescent state have been discussed. The item (7) concerns how shear flow affects formation of the dissipative structures in the semidilute solutions which are brought into two-phase region in quiescent state. The details were reported elsewhere [125].

### 6.6 Shear-Induced Dissipative Structures Formed for Semidilute Crystallizable Polymer Solutions

Up to this stage, discussion has focused only upon shear-induced structures formed for semidilute noncrystallizable polymer solutions. What would happen if polymers are crystallizable? This case involves intriguing interplays of two kinds of phase transitions on the formation of shear-induced dissipative structures, *shear-induced liquid–liquid (L-L) phase transition* (between a single-phase solution and a phase-separated solution) and *shear-induced liquid–solid phase transition* (crystallization and melting). The interplays generally depend on the following two characteristic temperatures, the phase separation temperature *T*
_{c}(0) (defined in *Figure*
*8-33*
) and the nominal melting temperature *T*
_{m}(0) for the quiescent semidilute solutions, as well as on shear rate.

We shall report below some results obtained for semidilute athermal solutions of ultra-high molecular weight polyethylene (UHMWPE) having *M*
_{w} = 2.0 × 10^{6} and *M*
_{w}/*M*
_{n} = 12 in paraffin wax as a solvent at temperature higher than melting point of the wax (69°C) [110, 113, 114]. The solution has polymer concentration of 5 wt% and *c*/*c** ≅ 11. Purification of the solutions is extremely important and indispensable for the experiments, because the refractive index increment \({\rm{(}}\partial n/\partial c{\rm{)}}_T\) of the solutions is very small compared with that of the semidilute solutions of PS in DOP [110]. The solution has *T*
_{m}(0) = 118°C and *T*
_{c}(0) below *T*
_{m}(0). At *T* < *T*
_{m}(0), the crystallization occurs even without shear flow, and hence the shear-induced phase separation concurrently occurs with crystallization so that the dissipative structure formation becomes quite complex.

*T*>

*T*

_{m}(0) so that crystallization occurs only under shear flow. The results obtained under this condition are summarized in

*Figure*

*8-48*. The part (a) summarizes shear-induced structures under continuous shear flow at steady-state as a function of \(\dot \gamma\). The behavior is apparently similar to that observed for the noncrystallizable semidilute solutions discussed earlier, except for some features as will be discussed later. The critical value \(\dot \gamma _{{\rm{c}},x}\) is the shear rate above which the shear-induced concentration fluctuations are built-up along the flow direction and hence where the butterfly pattern appears. The critical value \(\dot \gamma _{\rm{a}}\) is the one above which the optically anisotropic fibrillar (string-like) structures are formed, and hence the scattering pattern above \(\dot \gamma _{\rm a}\) is composed of the butterfly pattern plus the strong streak-like pattern oriented along the

*q*

_{ z }axis. \(\dot \gamma _{{\rm{c}},x}\) and \(\dot \gamma _{\rm{a}}\) are generally functions of

*T*. If

*T*≫

*T*

_{m}(0), for example

*T*= 150°C, \(\dot \gamma _{\rm{a}}\) exceeds the shear rate range accessible with the present experimental method \((\dot \gamma < 1,000\;{\rm{s}}^{ - {\rm{1}}} )\), so that only the butterfly pattern and shear-induced concentration fluctuations are observed with increasing \(\dot \gamma\) [110]. However, if

*T*is sufficiently low but yet above T

_{m}(0), for example at 124°C, the systems first show the shear-induced concentration fluctuation and then shear-induced anisotropic fibrillar (string-like) structures with increasing \(\dot \gamma\) above \(\dot \gamma_{\rm a}\) [113] as summarized in

*Figure*

*8-48a*.

The behaviors different from the noncrystallizable solutions are as follows [113]: (1) In the case of the crystallizable solutions, \(\dot \gamma _{{\rm cz}}\) could not be easily identified. This may be probably because as soon as \(\dot \gamma\) exceeds the critical shear rate \(\dot \gamma _{{\rm{c}},z}\) above which the scattered intensity along *q*
_{
z
} starts to rise up and hence demixed domains seemingly starts to be developed, the demixed domains are rapidly transformed into the optically anisotropic fibrils (strings) so that the gap between \(\dot \gamma _{{\rm{c}},z}\) and \(\gamma _a ,\;{\rm{{\rm \Delta} }}\dot \gamma \equiv \dot \gamma _{\rm{a}} - \dot \gamma _{{\rm{c}},z}\), is very small, if \(\dot \gamma _{{\rm cz}}\) exists or if not, \(\dot \gamma _{{\rm cz}}\) and \(\dot \gamma _{{\rm a}}\) are degenerated; (2) The anisotropic fibrils formed in the crystallizable solutions remained for a long time even after cessation of shear, while those formed in the noncrystallizable solutions decayed rapidly after the cessation, strongly implying that the *anisotropic fibrils* formed in the crystallizable polymer solution are a consequence of the *shear-induced crystallization*. It turns out that *T* = 124°C is close to the equilibrium melting temperature of the solutions in the quiescent state [113]. In any cases it is very important to stress here that the *shear-induced concentration fluctuations precede and trigger the shear-induced crystallization.*

*Figure*
*8-48b*
summarizes transient shear-induced structure formation for the case of: (b-1) \(\dot \gamma _{{\rm{c}},x} < \dot \gamma < \dot \gamma _{\rm{a}}\) where the shear-induced concentration fluctuations are formed and the butterfly pattern appears after the incubation time of *t*
_{c,x
}; (b-2) \(\dot \gamma > \dot \gamma _{\rm{a}}\) where, after the incubation period (*t* > *t*
_{c.x
}), the systems first form the concentration fluctuations (*t*
_{c,x
} < *t* < *t*
_{cz
}), then demixed domains through L-L phase separation (*t*
_{c,z
} < *t* < *t*
_{c,a}), and eventually the anisotropic fibrils (strings) (*t* > *t*
_{c,a}) [114]. The features described above also are apparently similar to those observed for the noncrystallizable solutions [33]. The L-L phase separation is not easily observed in the continuous shear flow experiments with increasing \(\dot \gamma\) (part *a*), though it is easily observed in the step-up shear experiments with increasing t (part *b*-2). This fact manifests itself a very narrow shear-rate range between \(\dot \gamma_{c,z}\) and \(\dot \gamma_{\rm a}\), if \(\dot \gamma_{c,z}\) exists. Therefore \(\dot \gamma_{c,z}\) may be observed with a sufficiently small increment of \(\dot \gamma\) across \(\dot \gamma_{\rm a}\). (2) The anisotropic fibrils formed in the case of (b-2) are again a consequence of the shear-induced crystallization triggered by the shear-induced L-L phase separation.

What would be the role of the shear-induced phase separation in triggering the shear-induced crystallization? This is an important problem to be solved in future works. However the following speculation may be offered at this stage. The demixed domains, which are aligned in the string-like assembly parallel to the flow direction, may undergo diffusion along the string axis, resulting in coalescence and elastic recoil of the domains. This coalescence-recoil process may enhance to create chains interconnecting the domains and to stretch the thus formed interconnecting chains into bundles of oriented chains. Hence the strings themselves are expected to enhance formation and growth of the bundles of stretched chains interconnecting the demixed domains.

Finally, it is important to point out a series of important earlier works along the line discussed here in this section, specifically on the rheo-optical works reported by McHugh and his coworkers [126–129]. The works are primarily devoted to elucidation of the shear-induced liquid-liquid “precursor formation” (phase separation) and subsequent crystallization into shishi-kebab formation for *dilute crystallizable polymer solutions* in tubular flow geometry. Although both systems (studied by Hashimoto and coworkers and by McHugh and coworkers) have common characteristics in terms of the dynamical asymmetry of constituent elements, the ways on which the elastic effects affect shear-induced phase separation and crystallization should be quite different between the two systems. In more general terms, the nonequilibrium pathways, by which the dissipative structures are formed, may be very different. This may become quite obvious when one imagines differences in the stress level and in the level of the concentration fluctuations built-up for the two systems under shear flow prior to the crystallization. Clarifications of the differences between the two systems in the shear-induced structure formation process and mechanism would be crucial and well-deserving of future investigation.

At this stage, a general remark applicable to this chapter as a whole is noted, independently of the content of the particular section under consideration. This chapter intends to give a theoretical interpretation of the shear-induced phase transition or shear-induced changes in the critical temperatures (drop (*Section*
*5.5*
) or up (*Section*
*6.1*
)) on the basis of the basic kinetic equation (e.g., HFMO theory introduced in the beginning of this chapter). However, it should be pointed out that there is the thermodynamic approach also. This approach phenomenologically incorporates the effects of steady flow on the systems by adding a stored energy term to the Gibbs free energy of mixing in order to account for the shear-induced mixing and demixing [86–94]. It is natural, however, that the thermodynamic approach does not predict the dynamics of the phase transition process and the shear-induced structures.

## Acknowledgements

The author gratefully acknowledges prof. A. Onuki for his collaboration, enlightening comments, and discussion on the subject of this chapter. The author is deeply grateful for professors J.S. Higgins and J. Mewis who have read the text and given valuable comments. The author thanks also professors C.C. Han and H.H. Winter for many comments on this chapter.

### Glossary of Symbols

Parameter characterizing for dynamical asymmetry defined by eq. 6

*c/c**

Reduced polymer concentration for polymer solutions

*c**

Overlap polymer concentration for polymer solutions

*C*

_{g}

A constant related to the gradient free energy arising from the nonlocality of interactions

Cahn–Hilliard–Cook

Degree of polymerization

Dioctylphthalate

Local concentration fluctuations of polymer for polymer solutions or one component of polymer in binary polymer mixtures at position **r** and time *t*

*D*

_{app}

Collective diffusivity for binary mixtures

*D*

_{ K }(

*K*= A or B)

Self-diffusivity of *K*-th component.

Quench depth at \(\dot \gamma = 0,{\rm{ }}{\rm \Delta} {T}(0){ = T}_{\rm c} - {T}\) with *T*
_{c} being critical temperature at \(\dot \gamma\)= 0

Shear viscosity

Zero-shear viscosity

Fast Fourier transform

Ginzburg–Landau

Interfacial tension

Shear strain

Amplitude of oscillatory shear strain

Shear rate

Critical shear rate above which shear-induced single-phase formation occurs for binary mixtures

Critical shear rate for shear-induced concentration fluctuations above which scattered intensity along the flow direction (the *x* axis) starts to increase

Critical shear rate for shear-induced demixing above which scattered intensity along the neutral axis (the *z* axis) starts to increase

Critical shear rate for shear-induced demixing systems above which anomalies are observed in both scattering and rheological properties

Relaxation rate for concentration fluctuations in binary mixtures

Relaxation rate for deformation of polymer chains

Relaxation rate for disentanglement in entangled polymer systems

Mesh size of entangled polymer networks

Thermal correlation length

Average value of \(\xi\)

Viscoelastic length defined in eq. 13

Correlation length of string-like domain structures perpendicular to the string axis

Correlation length for thermal concentration fluctuation along the neutral axis (the *z* axis)

*I*(

*q*,

*t*)

Scattering intensity distribution with *q* at time t after onset of phase separation

*k*

_{B}

Boltzmann constant

Large amplitude oscillatory shear flow (strain)

Light scattering

Laser scanning confocal microscopy

*q*-Dependent Onsager kinetic coefficient

\(\Lambda (q)\) at \(q \to 0\)

Characteristic length of systems undergoing spinodal decomposition in the early stage spinodal decomposition which characterizes Fourier modes of fluctuations having a maximum growth rate

*M*

_{W},

*M*

_{N}

Weight and number average molecular weight, respectively

*N*

_{ K }(

*K*= A or B)

Degree of polymerization of *K*-th component

*N*

_{1}

The first normal stress difference

(transmission) Optical microscope

Ornstein–Zernicke equation (eqs. 28 and 29)

Angular frequency in oscillatory shear strain

Polybutadiene

Polyisoprene

Polystyrene

Local concentration of polymer in polymer solutions or one polymer component in binary polymer mixtures at position **r** and time *t*

*K*= A or B)

local composition of *K*-th component in binary mixtures

Average of \(\phi ({\rm{\bf r}},t)\) or average of \(\phi _K ({\rm{\bf r}},t)\) for one of the components (*K* = A or B) in binary mixtures

Strain phase in oscillatory shear strain

**q**

Scattering vector or wave vector for Fourier modes

*q*

Magnitude of **q**, *q* = \((4{\rm{\pi }}/\lambda )\sin (\theta /2)\) (\(\lambda\) and \(\theta\) being wavelength of incident beam and scattering angle in a scattering medium)

*q*

_{ x },

*q*

_{ y },

*q*

_{ z }

Component of **q** along flow direction, velocity gradient direction, and neutral direction, respectively

*q*

_{m}

*q* at a scattering maximum

*q*

_{m}(0)

Characteristic wave number which characterizes the Fourier modes of composition (or concentration) fluctuations having a maximum growth rate in early stage of spinodal decomposition

*r*

Length scale of observation

*R*

_{g}

Radius of gyration of polymer

Small-angle light scattering

Random copolymer of styrene and butadiene, poly(styrene-*r*-butadiene)

Spinodal decomposition

Squared Lorentzian function (eqs. 25 and 26) predicted by Debye–Bueche theory [68]

SALS under shear flow

Transmission optical microscopy under shear flow

Coefficient of the first normal stress difference

Shear stress tensor

Shear stress

Scattering angle

*t*

_{c}(\({\rm \Delta} {T}\))

Characteristic time of mixtures at quench depth \({\rm \Delta} {T}(0)\)

Characteristic time in binary mixtures as defined by \(\tau = [q_{\rm{m}} ^2 (0)/D_{{\rm{app}}} ]^{ - 1}\)

\(\tau\) for polymer mixtures

\(\tau\) for small-molecule mixtures

*T*

Absolute temperature

*T*

_{c}(0)

Critical temperature of mixtures at \(\dot \gamma = 0\)

*T*

_{c}(\(\dot \gamma\))

Critical temperature of mixtures under shear flow at \(\dot \gamma \ne 0\)

*T*

_{cl}

Cloud-point temperature

*t*

_{I}

Characteristic interface thickness

Time-dependent Ginzburg–Landau equation (theory)